فشرف بي لشتغالي بالعلم ولا تبغي به ما عشت يا دا دا دا لا وياله من شراف عظيمي الحمد لله رب العالمين له الحمد الحسن والثناء الجميل واشدوا الله إلا الله وحده لا شريك له يقول الحق وهو يهدي سبيل واشدوا النسيد ونبينا محمد صلى الله عليه وعلى آله وأصحابه والتابعين لهم بإحسانين إلى يوم الدين أما بعد نحن في تفسير صورة النبأ ونوقفت في قائد 23 الله سبحانه وتعالى يقول لا بثين they will stay in the hellfire فيها احقابا ونعلم الثلاثة أقوال الثلاثة أقوال بحيث السكولرين في ماذا يعني by the word احقاب ماذا يعني بحيث الثلاثة أقوال الثلاثة أقوال الثلاثة but I want to insha'Allah وإتعالى mention the Adilla the evidences other than this evidence that has come to show that the disbelievers are going to be in the hellfire forever and they will never come out الله سبحانه وتعالى يقول صورة الحجر ayah 48 الله says وما هم and they are not منها بمخرجين the disbelievers are not those who will be taken out of the hellfire once they enter the hellfire الله سبحانه وتعالى He will never take them out in surah to such that Allah says وما اللذين فسقوا the word فسقوا هي means كفار it doesn't mean the mu'minin فما أواهم النار the hellfire will be the final abode كلما أرادوا أن يخرجوا منها whenever they wish to come out every time they want to come out Allah says in his verse وعيدوا فيها they will be brought back into it so they will always be brought back into the hellfire and the word كلما في العربك language it shows a تقرار that whenever they try to come out or try to even come out they will be brought back into it الله سبحانه وتعالى He says in another ayah إنه من يأتي ربه مجرما anyone who comes to Allah a criminal here again it means the disbelievers فإن له جهنما for verily that person جهنما is his abode لا يموت فيها ولا يحيا he will not live in it nor will he die so look at this it's a stage between life and death they're not really receiving a real life nor are they actually dying because you can imagine when a person reaches a point in their life where they want to die and the people are going to wish to die that day الله also says in another ayah كلما نضيجت جلودهم بدلناهم جلودا غيرها ليذوق العداب every time their skill finishes from the severe punishment of the hellfire and the heat of jahannam الله سبحانه وتعالى He says to us بدلناهم جلودا غيرها ليذوق العداب that Allah سبحانه وتعالى He gives them a new set of skin so they can taste the pain of the hellfire so it will never stop so continue a cycle for them also the prophet صلى الله عليه وسلم he told us in a hadith المسلم من ريطن من حريث أبي سعيدن الخضرير ربي الله تعالى عنه مرفوعا that it will be set to the people of jannah خلود فلا موت oh you people in the hellfire خلود this is eternity فلا موت there is no death for you ويا أهلا الناري oh people of jannah people of nahr خلود this is your final abode this is eternity for you and there is no death فلا موت عائشة said if anybody that day could die out of happiness the people of jannah would and if a people could die that day out of distress and depression the people hellfire would there is no way out of this this is what they are going to be for the rest of whatever is going to be there's no time it's continuous eternity it will never stop ولي ذلك أهل السنة والجباعة I believe that the خلود for the disbelievers is continuous يريدون أن يخرجوا من الناري وما هم بخارجين منها ولهم عذاب مقيم they want to come out of the hellfire they desire to come out of the hellfire الله سبحانه وتعالى يسيث وما هم بخارجين منها they will never be taken out of the hellfire and Allah says for them ولهم أن it is for them عذابون a punishment مقيمون that will forever carry on so if you've been in the hellfire doesn't mean that the punishment stops even the punishment is also continuous it's just that every time they receive a form of punishment then another form of punishment will start and then another form of punishment each time they are receiving a new form of punishment may Allah protect us from it ولي ذلك أهل السنة والجباعة if a person wants to save a person they should save them from كفر and should save them from shirk this is something that Allah doesn't forgive Allah does not forgive a person who comes with shirk إن الله لا يغفروا أن يشرك به ويغفروا ما يدونه دا لكلمة يشاء الله does not forgive a person who comes with shirk the day of judgment الله forgives any other sin that is below shirk so if you have a family member who is indulging into shirk or you know your family are not Muslims yet or you know a person who you can guide them the best gift that you can give to a person is to bring them into Islam and to save them from this punishment then Allah سبحانه وتعالى He says in this fire Allah is now going to describe it قال تعالى الله says لا يضوءون فيها they will not taste بردن ولا شرابة المفصرين the scholars of Tafsir they differed amongst themselves what is meant by the word بردن what does بردن mean there's two views the first view is بردن here means برد الريح there's no wind there's no air conditioner there's no breeze that they're going to receive the heat the hotness there is no breeze that they're going to receive and also they're not going to receive no sleep the برد here is a gnome that's the second view they're not going to be able to sleep because the عداب is going on for so long so the people of Hellfire there's no sleep for them the عداب is continuous the eyes will not close their mind cannot shut so Allah says لا يضوءون they are not going to taste فيها إن جهنم بردن ولا شرابا they're not going to receive برد which is the thing that they're looking for which is they're looking for something cool this cool here can either be a breeze a cool breeze or it could also be what sleep neither of the two are they going to receive ولا شرابا and they're also not going to be given anything to drink they are thirsty they want to drink they're in a place يا اخوة الكلام خطة than what you've ever seen the messenger told us عليه الصلاة والسلام that the heat that we receive when it's summer and we think to ourselves wow 50 degrees celsius in some countries in the world and you think it's very hot الله سبحانه وتعالى the Prophet ﷺ told us إن حر شمسي the heat of their son the Prophet ﷺ told us يزمين فيح جهنم it's from the breathing of jahannam جهنم أيت itself and he complained to Allah who ﷺ he said oh Allah this is my situation الله سبحانه وتعالى made for jahannam two times where it breathes one of its breathing is in summer where the heat is so hot the Prophet ﷺ told us this is what it is so that heat that we receive which is far from us جهنم they're in it they're burning in it and they want to drink if somebody set you in summer in that heat you're not allowed to drink anything you will dehydrate and die they're in it and they're not allowed to drink anything they're not going to receive any air conditioner no cold breeze this is their punishment الله then says for them سبحانه وتعالى the only thing that they're going to taste and the only thing that they're going to have go into their system is إلا حميماً وغساقا this is an exception this is the only thing two things the first one is حميماً حميماً is الماء الذي بلغ النهاية في حرارته water that has reached the pinnacle of hotness it's hot boiling water would be given to them they're thirsty they eat جهنم they want cold water to drink they're going to be given hot water this is their punishment وغساقا غساق is two things two goals of the Mufasiril the first one is they're going to be having the waste and the filth that has come from the people of Hellfire this is going to be added on to what they're going to drink and they're going to be told to drink so they're going to drink from the waste of the people of the Hellfire and the filth that has come from their body this is going to gather somewhere and it's going to be poured for them and it's going to be told for them to drink the other thing is they're going to drink the Hellfire جزم هريرا which is a punishment for Hell they're going to be poured with جزم هريرا and they're going to be told this is what you're going to have so there's no cold juice or drink that they're going to have now a person needs to ponder here needs to think why is this happening to these people is that Allah oppressing them سبحانه وتعالى is this Allah سبحانه وتعالى doing injustice towards them the answer to this question is what there's a box for astra s and z I think eight seven eight seven nine five maybe I think you're blocking someone or something so why is this happening to these people what's the reason Allah says to us the reason why these people are being punished like this they are adab is continuous they're going to be drinking this this is why they're going to be deprived from cold water what did they do wrong why what happened Allah says the reason I'm doing this is جزاة جزاة وفاقا what does جزاة وفاقا I mean an exact recompenses towards what according to their evil crimes the crimes that they did this is exactly the result of it they are only tasting the outcome of their actions they're the ones who did this to themselves الله سبحانه وتعالى it's only rewarding them based on what their action was و لذلك five of the great مفصلين they said ابن عباس مجاهد ابن جبرين قتالت ابن دعمة السلوسي الربيع و ابن زيد five of these scholars تفسير all five of them said ثوابا موافقا لأعمالهم this is in accordance to the action that they fulfilled they came with kufr and disbelief of Allah سبحانه وتعالى and now because they came with that الله سبحانه وتعالى is judging them based on what their hands have fulfilled so when you come the day of judgment and you're being sorry, since you're in this world today and you have this opportunity look at every action that you do و لذلك الله سبحانه وتعالى is going to tell us later what the believers are going to receive and insha'Allah وتعالى some matters between between that we'll expand on it later then الله سبحانه وتعالى goes on to saying إِنَّهُمْ دَيْوَا بَرَيْلِ لَا يَرُجُونَ حِسَارًا الله now explains more to us why He's punishing them and He doesn't have to but Allah chooses to do He says إِنَّهُمْ دَيْوَا كَانُوْ دَيْوَا they're used to لَا يَرُجُونَ حِسَارًا they're used to not look for a reckoning that's one view there's four opinions regarding what's meant by لَا يَرُجُونَ حِسَارًا number one the first view is لَا يَخافونَ حِسَارًا they never used to fear the punishment of Allah and that Allah is going to account them the fear was missing from them so the first reason why these people are being punished like this and they're being destroyed like this in the Hellfire number one is what إِنَّهُمْ كَانُوْ لَا يَرُجُونَ حِسَارًا it means اي لَا يَخافونَ حِسَارًا they were not scared okay the second view is the second goal is they were not hoping لَا يَرُجُونَ تَوَابَ الحِسَارًا the second is that they were not hoping for the reward of Allah سبحانه و تعالى that awaits them the day of judgment they didn't hope for that the third is the third view is لا يَرُجُونَ إِمِينَس اي لا يَتَوَقَعونَ they didn't even assume that there's going to be an accountability the day of judgment and that they're going to be accounted for what they said and did and last but not least إِنَّهُمْ لَا يُنُونَ بِالْبَعْتِ that they didn't even believe in the resurrection that they're going to be resurrected by Allah سبحانه و تعالى those are four statements of the great scholars of Tafsira إِنَّهُمْ دَيْوَا لَا يَرُجُونَ حِيْسَابَا they were never scared these tawagheet tawut is every single person who exceeds his limits that was sent for him كانوا في الدنيا whilst they were in this world لا يخافون that they never were scared and you jaziahum أحد they were never actually scared that they will be accounted for what they came with their evil actions so here it is that they're tasting their punishment so what else الله سبحانه و تعالى after that و قدبوا بأياتنا كدابة but they denied our ayat the universal signs that we sent for them the sun, the moon and everything we mentioned before they all denied everything and they also denied the verses the Quran was read on them they whilst they were in this world their parents their uncles their friends everybody was telling them في الله change your ways the way you're living is not right you need to rectify your actions you need to rectify your speech why are you living a life like this و قدبوا بأياتنا كدابة they disbelieved in it they didn't take those statements serious الله gave them many signs they didn't take those signs serious what were those signs that Allah gave them their friend died their other friend died and their other friend died and when an angel of death thinks those around you it's a sign that he's coming for you it's him telling you get ready I'm going to come for you soon but all of those signs they disbelieved in it و قدبوا they disbelieved بأياتنا our verses كدابة then a click on disbelief so this is and they didn't believe in the Quran they looked at the Quran as when I grow older and I become big I will change but an angel of death doesn't really look at that does it the day it came and it knocked on their door he didn't ask them how old are you were you young it took them و قدبوا بأياتنا كدابة but the Dalekan brothers I say the person who believes in these verses who truly means a lot to them and they actually believe in it it's not just a statement they will say it will change their life they will live by it you'll see it on their face you'll see it on their limbs you'll see it in their actions you'll see their life just fully changed a person who believes in these verses today was Juma'a he wouldn't have just been at home he would have been in the mosque crying asking Allah there's a time within the Juma'a if a person asks Allah Allah accepts his dua a person who believes in these verses these are the things that he's going to come with the messenger told us عليه الصلاة والسلام that the person who recites آيات القرسي after every prayer the only thing that he awaits for entering جنة is death all he has to wait for is death and then as soon as that he will enter جنة so the one who believes in these verses is going to take all of that series and his life and his actions are going to be changed if you don't believe in them you're going to come with nothing وكذبوا they disbelieved بآياتنا عوة برسي كذبا they disbelieve so they didn't believe what the Qur'an came with and Allah then says وكل شيء in an everything احصينه كتابة everything that they've done we recorded it in a book every statement you say يا عبد الله يا أمت الله the slave of Allah male or female every single statement that you have ever said that you forgot الله hasn't forgotten ما يلفض من قول him إلا لديه رقيبا عتيد everything statement that you've said that you forgot الله to him is documented هذا كتابنا ينطق عليكم بالحق إننا كنا نستنسكم ما كنتم تعملون this is our book every single thing that you used to do we documented it for you so you don't disbelieve it this was the date that you said it this is the place that you did it everything will be placed in front of you الله says in another ayah إن رسولنا our messengers our angels إن رسولنا our messengers they were writing يكتبون ما تم قرون the plotting and the planning and the evil things that you were doing behind closed doors in the dark الله says all of that our angels were writing it and they were documenting everything you were doing الله says in another ayah وكل إنسان ألزمناه طائره في عنقه ونخبت له يوم القيامة كتابا يلقاه من شورا إقرأ كتابك كفى بنفسك اليوم عليك حسيمة that day your book will be brought out in front of you and it will be set to you إقرأ كتابك read your book or slave كفى بنفسك اليوم عليك حسيمة this book is sufficient to tell for you to busy yourself with this is it it will take all your time because it has every single thing as الله سلنناه هذا آية ووضع الكتاب فترى المجرمين مشفقين من ما فيه ويقولون يا ويلةنا ما عني هذا الكتاب لا يغادر صغيرة ولا كبيرة إلا أحصاها ووجدوا ما عملوا حاضرة ولا يضلم ربكة حدا everything they did the little the big the small everything you ever said and did it's documented it's right in front of you what is this book what is this book that has left has not left anything behind that is documented all my life الله تعالى سبحانه وتعالى الله already knows but you know what the powerful thing here is it's not just الله is judging you based on his knowledge الله is also based judging you based on what you're going to read your documents it's in front of you and remember الله سبحانه وتعالى he is not one you can deceive today if you go to a court of law if you go to a court you could get a good lawyer and you can get away with some big crimes that you've done if you if your back pocket is swollen and you got money and you can get and you got the right links and you can get the best lawyers you might walk away in that courtroom being an innocent person and blaming blaming all the the crimes on somebody else but the day of judgment الله سبحانه وتعالى he knows every single thing so my beloved brothers and sisters ponder here and think ask yourself when you fall into the sins that you fall into and you do these crimes that you do ask yourself you wouldn't even say that in front of your your parents and you wouldn't say that in front of you know a righteous person in the family why would you say that in front of الله سبحانه وتعالى why would you do that in front of your Lord why would you do this and say this in front of the angels why are you going to make الله the lowest of those who look at you وكل شيء in everything the word كل in Arabic language is from the most generic words that show generalization with me brothers ابلغ العمومات it's the most powerful when it comes to everything كل شيء everything احصيناه كتابة we have documented it not a split not a لحظة split second of your life is not a document for us everything ولي ذلك some of the علمات they say ما يلفض من قولين إلا لديه رقيب ونعتين the good and the bad and what about the things that are not good and the bad even they're documented because of the generalization that's used here even the things that are not wrong that you said are still documented everything is documented that day الله سبحانه وتعالى so here there's an issue here brothers which is in the next ayah which is الله سبحانه وتعالى he says فضوقوا فلنزيدكم إلا عدامة الله سبحانه وتعالى here he says الله سبحانه وتعالى here he says فضوقوا it will be said to them فضوقوا taste here the verse says فضوقوا so taste فلنزيدكم we will not increase you إلا عدام and accept punishment and accept torment this is what's going to be said to them here the Mufassireen scholars of Tafseer they discussed this verse because this verse shows that the people of the half eye will be spoken to and Allah says in another ayah إن الذين يشترون بأحد الله ثمن قليلا وإيمان إن ثمن قليلا those who have bought the covenant of Allah and the oath that they took with الله سبحانه وتعالى they bought it with ثمن قليلا some small money الله says أولئك لا خلاق لهم those individuals they have no خلاق meaning they have nothing with their Lord ولا يكلموهم الله الله will not speak to them that day ولا ينظروا إليهم أن الله will not look at them يوم القيامة ولا يزكيم الله will not purify them ولا هم عذابون أليم and they have severe punishment so this verse clearly says ولا يكلموهم الله الله will not speak to them and the verse that's in front of us here الله is saying فذوقو تيست so it seems that Allah is speaking to them سبحانه وتعالى so how do we reconcile between these two verses one verse is saying الله does not speak to these people and another verse الله is saying فذوقو تيست فلنزيدكم إلا عذابا they took two answers in which they responded to this verse the first one is ولا يكلموهم الله means الله will not speak to them a speech that will benefit them something that will save them from the punishment of the Hellfire will not be told to them the speech that's being negated here is not unrestricted it's a particular type of speech that they won't receive the speech that they're not going to get is الله will not speak to them a speech that will benefit them every speech that will be said to them is going to hurt them more the speeches are going to be a form of punishment for them they're not going to receive any kind words from their Lord سبحانه وتعالى that's one statement the other statement is that when Allah says in this ayah ولا يكلموهم الله will not speak to them أن مواقف القيامة تتعدد that the day of judgment the stages are different so some places الله will not speak to them and in some other places He will speak to them سبحانه وتعالى so those are the two responses that the scholars have given رحمهم الله جميعا this is what's going to be said to them فضوقوا تيس فلنزيدكم we will not increase in for you إلا عداباً except a punishment if you look at the Quran and you ponder on the Quran you always find الله talks about the people of the Hellfire and the punishment that awaits the people of the Hellfire and what does he always do after that he mentioned the people of righteousness the righteous slaves of Allah the متقين the people who are obedient to him and following his command الله always also mentions them and this is something you can always see in the Quran الله mentions the kuffar الله mentions the مجرمين and then Allah goes to the متقين and the مؤمنين so Allah says سبحانه وتعالى now إن للمتقين مفازا الله moves on to the متقين He says سبحانه وتعالى إن للمتقين مفازا حداق وعنابا إن للمتقين for the متقين those who have تقوى there will be a success إن للمتقين مفازا for them will be success here the word مفازا the scholars have explained it with two explanations some scholars have said unrestricted success so they said مطلق الفوز unrestricted success which basically is they are going to receive everything they've ever wanted and they're also going to be prevented from every evil that they ever wanted to stay away from stay away from and they didn't want to come to them it won't the second view of the مفاصرين is إن للمتقين مفازا and here specifically means that the مفاز specifically means جنة that's it not unrestricted for success specifically just means جنة so the متقين for them is مفازا and if you look at the two goals they come together because in جنة there's unrestricted success مالعين رأت ولا أذن سمعت ولا خطر على قلبي بشر eyes have never seen it ears have never heard of it and it has never come to the heart of any believer what's in جنة and now you're going to see powerful descriptions for جنة what does Allah want from us brothers this is the thing Allah wants to be scared and Allah now wants you to have hope all of the things that were mentioned about the people of the Hellfire and the punishment that they go to go through it's fear that's needed from you and now hope hope for جنة and to be from the متقين because if Allah only tells you about the disbelievers and the punishment of the Hellfire you might give up in life that's what Allah has said in the Quran نبع عبادي أني أنا الغفور الرحيم وأن عذابي هو العذاب الأليم غافر الظنبي وقابل التوبي شديد العقابل التوبي الله always mentions it like that the Quran فيا غافر الظم and also غافر الظنبي وقابل التوبي شديد العقاب he's غافر he forgives and he's once a punishment he'll punish you so Allah سبحانه وتعالى these two winds is what a bird needs to fly with the wing of hope and the wing of fear with those two you can fly if you try to worship Allah only on fear then you take the part of the khawarij and if you only worship Allah based on the part of hope then you've taken the part of the murjah and if you worship Allah only on love you've taken the part of the Sufiya you need to worship Allah with those three الخوفور رجع والمحب fear hope and love all together when it comes together that's what the mu'minina like not one over comes the other that's why Allah now is telling us إنة للمتقينة وفائزة for the متقين إخواني what is a متقين a متقين where does this person come keep him with his characteristics بلق من حبيب رحمه الله he gave the best definition of taqwa his definition of taqwa is does anyone know the definition that بلق من حبيب gave رحمه الله two woods taqwa because now we speak about taqwa some of you may not even know what it means and if you don't know what taqwa means how are you going to attain what you don't understand does anyone know the definition this is the best definition that has been given regarding a taqwa I think we should make this a homework then next تفسير لسل everybody tried to look up طلق من حبيب what he said regarding a taqwa it's important that is the best definition that has been given regarding a taqwa it is a comprehensive definition his name is طلق إبن حبيب رحمه الله he gave the best definition إنة للمتقينة وفائزة for the people of taqwa is success by attaining جنة by being distance from the house fire


In this lesson we're going to dig a little bit deeper about how to find multiplicative inverses in a more efficient way than what we saw in the last lesson. So computing greatest common divisors in multiplicative inverses isn't really that hard from a mathematical point of view, but it does require factoring numbers down, which takes a long time. And this is the first time we're encountering that in this course, but it is a big idea in modern cryptography, so we'll see it time and time again throughout the course. Computing is a slow process, and it only gets slower as numbers get bigger and bigger. So if a number is twice as large, it might actually take more than twice the time to factor that new number. Luckily, there's some nice mathematics that have made factoring a little bit faster than it would be otherwise. And this is a piece of mathematics that comes from Euclid, Euclid that you've maybe have heard about in a geometry class, more specifically Euclid of Alexandria, who helped develop the Euclidean algorithm, or as maybe better put is known for the Euclidean algorithm. It appeared in his book called Euclid's Elements back around 300 BC, and it can really efficiently compute the greatest common divisor of two numbers, which we'll see is really going to be helpful for our work. In fact, this algorithm is probably not developed by Euclid. He was pretty, pretty well known for often just compiling work by other mathematicians that came before him. And since then, this algorithm was discovered independently from his work in both India and China, although it might have happened to be a century or two later, it's a little bit of a falsehood to say this algorithm truly belongs to Euclid. But what is important is that we know it's been around for over 2000 years, and it's still got some great uses today. So let's see this algorithm in action, and then we'll dig in at why it works. So suppose we've got two numbers, A, which is 707, and B, which is 980, and you're interested in finding the greatest common divisor of those two numbers. Well, what the Euclidean algorithm claims is that instead of solving the original problem, which would involve factoring 707, factoring 980, and looking for all of the common factors to build up that common divisor, the biggest one, we could instead simplify the problem by instead finding the greatest common divisor of 707, the first number, and 980 minus 707, the difference between them. Notice we just subtract the smaller from the biggest, but mathematically this would work either way, but this will get us to our result much faster. So now our problem is to find the greatest common divisor of 707 and 273. Smaller numbers are easier to factor, faster to factor. But why stop at this once? If that rule is true, and we've not proven that it is yet, why would we sell ourselves short? We should do the same trick again. Let's take that 273 and subtract it from 707, and now find the greatest common divisor between 434 and 273, even smaller numbers. But why stop now? Go again. And instead we can find the greatest common divisor between 161 and 273. And we can go back and forth like this, subtracting the smaller number from the largest number, until eventually we get down to either these two numbers or the same, in which case we can pretty easily inspect that the greatest common divisor of 7 and 7 is itself 7. Or if you went one step further, maybe you didn't catch that, one of your two terms would be 0, by which case you know the other number must be the greatest common divisor. So it turns out this is much faster than factoring 707 and 980 down to its component primes, grouping them up to get the greatest common divisor, because this is just a bunch of simple subtractions, which for both a human and a computer are very easy to do. So instead of thinking that goes on, just a process that you can follow. So they're much quicker to do. Let's figure out why this works, though. All hinges on the fact that Euclid poses that instead of finding the greatest common divisor of the two numbers as they are, that instead you can find the greatest common divisor of the smaller number, and then the difference between the bigger and the smaller number, and you'll get the same answer. It turns out you can check that it is true, 7 is the greatest common divisor of 707 and 980, but let's find out why. So let's talk about why the Euclidean algorithm works. Let's start by supposing we have some number n that we define to be the product of d and x. We have a second number m, which we define to be the product of d and y, where d, x, and y are all positive integers, and then therefore n and m are also product integers, positive integers. Notice that d will divide both the numbers n and m, because they both, n and m, have d as a factor. And that's how we define divisibility when we're talking about numbers, that if a number has a second number as one of its factors, that second number divides the first number. So d divides n, d divides m, because d is a factor of both of those numbers. And then one notational note, we're going to start using this notation, this vertical line that separates the two numbers d and n as shorthand for saying d divides n. So we'll start seeing that on the next slide, just wanted to prepare you. So one fact that'll be helpful for us to explain why the Euclidean algorithm works the way it does is that if we were to take the difference between those two numbers m and n, so substituting dy for m and dx for n, we could factor the d out, and we're left with a product of d and y minus x, which means that d is a factor of n minus n, because it's a product with another integer, which means that by our definition of divisibility, d is a number that will also divide the difference between m and n. So that might make sense, might not seem very useful yet, but that'll be a very important fact for where we're going next. So let's dig in and see why this Euclidean algorithm must work all of the time. We'll start by defining two things. That d is the greatest common divisor of x and y, and that d prime is the greatest common divider of x and the difference between y and x. Our goal is to prove that those two numbers are actually the same. And if that's true, well, then what Euclid showed us, the algorithm we saw previously must always work. It wasn't just a fluke. Here's how we're going to prove it. It's going to hinge on these two statements here, which maybe seem obvious, but let's reiterate one more time. If d is the greatest common divisor of x and y, it is the largest divisor of x and y. If d prime is the greatest common divisor of x and y minus x, d prime is the largest divisor of those two numbers. There could be other divisors, but if there are, they must be smaller. So let's use this all to our advantage here. Since d is the greatest common divisor of x and y, d must divide both x and y. And as we just saw on the previous slide, that means d must also divide the difference between y and x. So we currently know that d is a divisor of both x and y minus x. Well, we already know that d prime also is a divisor of x and y minus x, but d prime must be the largest divisor of those two numbers. That's why it's the greatest common divisor. So it's all great that d divides both of them too, but it just means that d must be smaller than d prime. Hold on to that thought. Let's work that same kind of logic on the second statement. Let's look at d prime. d prime divides x, d prime divides the difference between y and x. And with a little bit of reasoning, very similar to what we just saw on the last page, that if we were to add x and y minus x together, we could figure out that d prime also divides y. So if d prime divides x and y, just like d does, well, that just means d prime is a divisor of x and y. But it can't be the biggest common divisor between x and y. That prize is already taken by d. So good job, d prime. You divide both of them. But you just got to be a smaller than number than d. But let's look at these two statements we have. How could d both be less than or equal to d prime, but d prime be less than or equal to d? It must mean that the less than part really isn't the important part there. It must mean that those two things are equal. The only way to make both those inequalities that we have proven to be true simultaneously is if the value for d and d prime are, in fact, the same. And that's it. We've proven it. We have proven that the greatest common divisor between x and y and the greatest common divisor between x and y minus x must be the same number. So now we can use the Euclidean algorithm to find greatest common divisors without fear that it's ever going to go haywire and not give us a correct result. We've proven it to be true all of the time. So let's see how we can maybe apply this Euclidean algorithm in a more general sense, which might help us think about how we could program the Euclidean algorithm using Python. So instead of choosing specific values for a and b, let's keep them general and just say they're integers. And our goal now is going to find the greatest common divisor of a and b, which now we know we can just do the greatest common divisor of a and b minus a. So in Python, what we might choose to do is redefine b to be b minus a. So then we can just say, well, I'm going to find the greatest common divisor of a and this new value of b, which we'll find some way to do in code and just say, let's subtract the smaller number a from the bigger number b. And we'll do that for a while. We can repeat this step. Let's always subtract the smaller number of the two from the bigger one. Maybe sometimes it means we're going to have to redefine a, because if we do enough subtractions off of b, maybe b is now, in fact, the smaller one. So we'll need to subtract b from a and then redefine a and do it again and again and again until a and b end up being the exact same values, a is equal to b. And when that's true, since they're the same, we can just look at whatever a or b is. In this case, I've chosen a to figure out our greatest common divisor. So that is going to be our strategy here. We're going to compare a and b, subtract the smaller one from the bigger one and redefine it to where the bigger one used to be and then check again, which one's bigger. Subtract the small one from the big one, redefine that difference to be where the big one used to be and we're going to keep going and going and going, repeating, repeating, repeating. Sounds like a good case for a loop. We don't know how many times that we're going to have to do this, so it sounds like a good case to use a while loop and we're going to keep doing this process until we find that a is equal to b. So as long as a and b are different numbers, we're going to repeat these operations. Let's take those ideas and turn them into some code. So I've chosen two numbers a and b to be two in 26. And remember, we want to subtract the smaller one from the bigger one until those two numbers are the same. So to get us started, we'll get the header of our while loop here. So we'll say, while does a does not equal b, so basically if a and b are equal, we should stop. But if they're not equal, we should do what's in the loop. Now what did we say we want to do? We want to subtract the small one from the big one, but we don't know which one of those is right away. We have to do some sort of comparison. So we can say, if a is bigger than b, then subtract b from a and then redefine that to be the new value of a. Else, and this is a new Python command for us today, we haven't seen this, else subtract a from b and redefine that to b. What the else statement does is that if your if statement ends up being false, you can give it a secondary line of code to run when that's the condition. So in this case, if a is bigger, subtract b from a, otherwise subtract a from b, and then it's gonna go back up to the top of the loop and it's gonna keep going, going, going until a and b ultimately end up being equal to each other, which means the loop will stop and it will show us the value of a, which is two, which we know is the greatest common divisor of two in 26. Let's real quick look at the visualization of this in code and notice it's only gonna take 40 steps for it to do it, which sounds like a lot, but for a computer goes really quickly. All right, let's take a look at this code run. So we'll step through and define our values for a and b to be two and 26 and we jump into the loop. So it's gonna compare those two and as long as they're not equal, which they are not, it will run the code. So it's gonna go to this if statement next and see if a is bigger than b, that's not true. So we're gonna hop over line five and we're gonna go to the content of the else statement instead. So it's going to do b minus a and store that back to b. So we should see b decreased by two when we run that next line of code. And it does and it goes back up to the top of the loop and checks are they equal yet? Nope, they're not. So we're gonna keep on going. Is a bigger than b?  So we're gonna keep stepping through this code. We can see b continuing to be incrementing lower and lower until eventually it hits two. And now when we get to this while loop, it says while a is not equal to b, but they are equal. This is no longer a true statement. So we get out of the loop and we run the final line of code line nine, which will print to the screen our greatest common divisor. Take another look with slightly different numbers to see how it affects the runtime. So we can see here that a is a slightly bigger number now seven, all the rest of the code is exactly the same. And as we step through, we can see the same thing start at first. And if we increment until b is now smaller than a and now when we go to the next step, the if statement resolves to be true. So we're gonna run line five. So we're gonna subtract b from a and now a is the smaller one. So we're gonna run the else statement and now a is still smaller. We're on the else statement once more. B is now smaller. We'll run it again. And now the two are equal and it turns out we jump out of our loop and our greatest common divisor is one, which is true since a is prime. The biggest divisor it must have a 26 is one since b equals 26 does not have seven as a factor. That only took 25 steps even shorter. So certain conditions for a and b will cause us to run faster or slower, but we'll see it's not that many steps for the computer to do to get a greatest common divisor. That's the beauty of this algorithm. It operates very efficiently. We can see here that if we redo the example of 707 and 980 and find their greatest common divisor of seven, it only took 37 steps. It's actually less than two in 26 in order for the computer to do it. And if you put in really big numbers, so 765,873 and 3,945,934 only takes the computer 148 steps in order to do that to find that those two are actually relatively prime. They share no factors besides one. We'll continue to work with this algorithm throughout the course and these numbers will only get larger and larger as we do it. It turns out there's even a slightly more efficient way in the computer for us to do this. This is the way Euclid envisioned this algorithm, but we've been able to update it with modern times and operations that maybe Euclid was not quite aware of at the moment. So we can do a little bit better, but we'll save that for another lesson. Turns out there's an extension to this algorithm. We could even do better than just finding the greatest common divisor. If we just keep track, we do a little bookkeeping as we work through our algorithm. We can actually figure out not only what the greatest common divisor is, but if that greatest common divisor is one, we can actually find the multiplicative inverse, which is really why we care about this in this course, is that we're trying to find multiplicative inverses for our various sized alphabets. So what this extended Euclidean algorithm allows you to do is solve equations very efficiently that are in the form. The greatest common divisor between two numbers equals the first number times x plus the second number b times y. So we're gonna look for x's and y's that solve this equation and we'll see why that's important to us to finding multiplicative inverses in just a moment. When we can find that the greatest common divisor between our two numbers is one, and once you know x and y, you can mod by a or b in order to get the following equations or maybe more specifically the following equivalencies. You could see that one is going to be congruent to by mod a, because if I mod by a, if you look at our equation, well, they're in the middle of the screen, it says b y plus some multiple of a. So if I mod by a, the a x part is equivalent to a zero. It's just we're adding on some multiple of a, some multiple of our modulus. So it actually doesn't impact our congruent statement. Likewise, if we were to mod by b and get a congruent statement here, well, we're just left with a x because the original equation was a x plus b y, so multiple of b. So if we mod by b, those multiples are equivalent to adding zero once we get it down to our congruent statement. Why is that important? Well, if we know that two numbers are congruent to one in a particular modulus, then we know those two numbers are actually inverses of each other, multiplicative inverses of each other, which is what we're trying to do, find the multiplicative inverse in a particular modulus. Let's see how this all works out. Say we wanna find the multiplicative inverse of seven in mod 26. You might remember from a previous lesson what that is, but I'm not gonna tell you, see if you remember, and then we're gonna go ahead and figure it out. Our goal is to obtain some equation in the form of the greatest common divisor of a and b, such that it's equal to a x plus b y. So for these specific numbers, the greatest common divisor of seven and 26, we know is one, we just saw it, and we're looking for that one to be equal to seven x plus 26 y. So a in this case is our seven, our key, and 26 is our m or our modulus or the size of our alphabet. So this is the goal. We wanna get our equation to look like this so we can figure out what x and y are, and then once we get it into this form, we'll be able to find our inverse. Let's get to work. So to get there, we have to start with these equations, and they're gonna seem pretty obvious and maybe not very useful at first, but trust me, we're gonna put them to good use in just a moment. Equation one is just specifying that 26 is equal to seven times zero plus 26 times one. The second equation is that seven is equal to seven times one plus 26 times zero. While these aren't inherently helpful quite yet, they are somewhat in the form of what we're trying to get to. A number on the left is equal to seven times a multiple plus 26 times a multiple. And we're gonna keep rearranging this pair of equations and try and work our way to that equation form that we're hoping for, which is a one on the left, seven times a number plus 26 times a number on the right. And we can do that by subtracting off multiples of equation two from equation one. This is just like our Euclidean algorithm we saw earlier. So I could subtract off seven from 26, but I can see I could probably do that a few times. I'm gonna speed it up. I'm gonna subtract off three times the second equation from the first equation. So three times seven is 21. 21 subtracted from 26 gives me the five on the left-hand side of the equation and our third equation down here at the bottom.  And I'm gonna keep it in that form. I know that's the same thing as negative 21, but I'm gonna keep it in this form to help keep track of how many multiples of seven there are since that's the form of our goal equation. And we're gonna do the same thing with a 26. One times 26 minus three times zero times 26 is still just one times 26 on the right-hand side of the plus sign. So we now have a new equation. The left-hand side is five, it's not one, but five is closer to one than either seven or 26 was. And in fact, let's do the same thing again with these two new equations. We'll rename equation two to equation one and we'll name that new equation we just got to be our new equation two. And you see, we can do the same process again. I could subtract off just equation two from equation one. And when I do that seven minus five gives me two on the left, seven times one minus seven times negative three gives me seven times positive four. And then 26 times zero minus 26 times one gives me a plus 26 times negative one. And we can verify this is still correct. Our equation is still true. Two is equals to 28 plus negative 26. And again, it's not exactly the form that I want but the two on the left is getting me closer to that one equals. So let's do this one more time. Same step, new equations. We can subtract off two multiples of equation two from equation one. So five minus two times two gives me the one on the left. Ooh, that's what we're shooting for. We're getting close now. And on the right hand side of the equals if we subtract off seven times negative three and we subtract from that two times seven times four we'll get seven times negative 11. And on the right hand side, same thing with the multiples of 26 we'll obtain 26 times three. And we can verify that this is correct that the number is on the right hand side actually do add up to one. And we're at our goal. We've got our equation into the form that we were hoping for. Now why was that helpful? We got the solutions x is negative 11 and y is three. Let's use them. So remember we have one is equal to seven times negative 11 plus 26 times three. Our goal is to figure out what number seven could be multiplied by in order for it to equal or be congruent to one in mod 26. If we take this equation in mod by 26 that 26 times three is effectively adding by zero. Remember any number that you add multiples of 26 to in this case we've added three multiples of 26 are effectively adding by zero after we're done applying the mod operator in a mod of 26. So we're left with the statement that one is congruent to seven times negative 11 in mod 26. So seven and a negative 11 then meet our requirements to be multiplicative inverses. They multiply together to give you a congruency value of one in mod 26 we did it. Now conventionally we want our inverse keys to be between zero and 25 if we're working in mod 26. So we can just do our mod operator on negative 11 to find out its equivalent is 15 in mod 26. So we would say that seven and 15 are multiplicative inverses in mod 26. And we've now just seen how the extended Euclidean algorithm allows us to quickly compute that took a few steps. But we'll see that this is a nice way to do this because the numbers get larger and larger and larger. 26 possible attempts. Yeah, we could write a little loop to brute force this for us. But when our mod is in the hundreds or thousands of digits, we're not gonna wanna try them all. We won't be able to try them all. But this algorithm will still work relatively efficiently in order to get that same result. We'll learn how to program this algorithm a little bit later on in the course. But for now, we're gonna be doing this by hand. So it might be helpful to find a little shorthand so you don't have to keep writing down all of those computations, all of them. So the way we're gonna do that is we're gonna keep track of, you might have already picked up on this watching the work, all we really need to keep track of are those two numbers that represent the multiples of 26 and seven. So I'm gonna create a little table along the right-hand side here as we copy down the equations that we produced on the previous couple of slides. So far, I've got this first equation 26 and it has one multiple of 26 and zero multiples of seven in order to produce that total of 26. The second equation we have with seven was equal to 26 times zero plus seven times one. So I've marked those coefficients to 26 and seven as zero and one on the table on the right-hand side. Then we had five was equal to 26 times one plus seven times negative three. So I've written the five down the left-hand column and then the one in the negative three to keep track of the multiples of 26 and seven that are indicated in the two columns. And then two, negative one and four become the coefficients in the table for the next equation and then one, three, negative 11 on the last row. And we know that negative 11, the one under the seven column because we've modded by 26, that negative 11 is our inverse and that's how we've got to 15 being our multiplicative inverse. I think that working with these, just these coefficients on the right is gonna be a much quicker process for you. We don't really care about the 26 and the sevens in the equations, we just care about the coefficients that come along for the ride. And we can perform that same difference or subtraction between one row and the next without having to write everything down. I think that's a really speeded up for you. And that's our Euclidean algorithm, what it is, how to use it and why it always works. I think you'll find it to be a very efficient way for you to first verify that a particular number has a multiplicative inverse and a particular modulus and then use the extended version of the algorithm to figure out what that inverse is.


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Mr. District Attorney! Radio friends, this is Harrington. I'm speaking to you from the District Attorney's office. Two months have gone by since we captured Duke Orlando at the municipal airport. Two months in which the District Attorney and our entire staff have been working day and night piecing together the evidence we have collected against Duke Orlando and his restaurant right. The Ohio Kid, the Duke's torpedo, has applied us with much of the connecting links that we needed. He has turned state's evidence, which means he'll be a witness for the District Attorney. The District Attorney has been in court for the past five days, and this morning he was going to call the Ohio Kid to the stand. Kid, I want to remind you that you have sworn to tell the truth while in the witness chair. Are all the statements you have made so far the truth, the things you described actually took place as you said? Yes, they did. You heard Mr. Tony, the restaurant owner on the witness stand, say that he was threatened by Lucky Lynch and yourself. Had Lucky said if he didn't join your employer's protective society, something real bad was going to happen to him or to his family. Yeah, Lucky said that. Did Duke Orlando give him orders to say that? Yeah, the Duke said if any restaurant owner don't want to join, just show them how badly they need protection. And how did you show Mr. Tony? Well, you know, you heard Mr. Tony tell how we bombed his house. Who's we? Lucky Lynch and me. Who told you to bomb Mr. Tony's house? The Duke. Was that the only time the Duke ordered you to do something like that for him? No, he was the boss of the whole racket. He gave us the orders on every job we pulled, the BB killing, the two-law murder. Order! Order in the court! Well, radio friends, the case of the people versus Duke Orlando is in its final moments. While he was on the witness stand, the Ohio kid disclosed the entire inside details of Orlando's restaurant racket. Now it looks as though we have a closed case against him. The district attorney is in court now making a summation to the jury. And so, gentlemen of the jury, there can be no doubt in your minds that Duke Orlando is guilty of the crime as charged. Look back on the facts that have been disclosed here in court. We have proved that Duke Orlando ruthlessly ordered the bombing of Mr. Tony's house, knowing that a little seven-year-old innocent child would be killed or maimed. We have proved that he coolly planned the death and disappearance of BB, the union official, that he carefully engineered a scheme to get rid of his former right-hand man Lucky Lynch, that he, but it's not necessary for me to go on. I know as intelligent citizens of this community that you will not hesitate for one moment, that you will render a verdict of murder in the first degree. Duke Orlando is as guilty of the murder of Professor Toulon as if he actually wielded the blow. Harrington speaking. No, the DA is still in court. Court recessed about five minutes ago. Well, yes, I expect them any minute. Well, I doubt if you'll have time for that today. Right. Goodbye. Oh, Harrington. Well, how'd it go, DA? Well, I think we've got the case clinched. And of course, there is still a summation for the defense. However, I'm not worried if Lester were still defending the Duke, I wouldn't be quite as sure of the outcome. Lester might get around the tremendous amount of evidence we brought out, but I don't think Stanton, his new attorney, will be able to. I'd like to, just to get Tony speaking. He does. That's slightly irregular. Well, all right, I don't see any harm. I'll be down to see him shortly. Harrington, Duke Orlando wants to see me. Don't tell me he wants to come clean now. I don't know, but I'll find out soon enough. I've got some work to clean up, but as soon as I'm through, I'll be paying Duke Orlando a visit. Hello, DA. They like my new quarters. Not quite as nice as your own place, Duke. Ah, they even got springs in the bed. You won't be here long enough to worry much about that. Trials pretty near over, ain't it, DA? Yes, but what did you want to see me about? I decided to give you a break. That's nice of you, Orlando. Yeah, I thought so. I'm going to confess. Why? Well, I don't think my mouthpiece doing so good. If you don't do something fast, we're going to lose this case. I don't think he has my interest at heart. You don't, eh? No, so I thought I'd confess, then you could let me off easy. What gave you that idea? Well, it saved you a lot of trouble. It would have before this trial started. I gave you your chance when Lucky Lynch died. You could have given up then. It's too late now, Duke. I've got you dead to rights. You're going to the electric chair. Still dreaming, DA, huh? Well, too bad you won't take me up on this confession idea. Have made it very easy for you, and you could have let me off easy. Nothing can make it easy for you now, Duke. It certainly won't be easy for you to listen to that jury's verdict. Gentlemen of the jury, have you reached a verdict? We have, Your Honor. What is that verdict? We find the defendant, Duke Orlando, guilty of the crime of murder and the first degree is charged. Duke Orlando, have you any legal cause, why judgment and sentence should not be passed upon you at this time? No, Your Honor. The sentence of the court, Duke Orlando, is that you should be confined to State's prison, and that on some day in the week beginning, June 11th, you'll be put to death in the mode and manner prescribed by the laws of this State. Electric chair. No. No. Hey, sit down, Duke. You've been walking up and down for an hour. It gets me nervous. Sit down, will you? I can't sit, I can't. Every time I sit, I think of a chair. It ain't so easy being quiet with that on your mind. Yeah, I guess you're right. Plenty of dark in these cells, ain't it, Duke? Funny they put you in a cell right next to mine. I won't be here long, kid. But you're here for life. Life in a cell is a long time. Life in a cell, Duke, ain't nearly as long as the time you're dead. Yeah, I guess you're right, kid. Duke, will you talk with me a couple of minutes? Maybe I'm getting soft, I don't know. Yeah, I ain't feeling so good myself. I'll even go further.   Just a lot of air that don't last. I never thought I'd hear you say something like that, Duke. Well, I never did neither, but this is different sitting here and thinking, you scared, kid? Yeah. Hey, Duke, did you ever fall for a nice dame, a regular dame? Once. What happened? She threw me over when she found out who I was. Do you remember one? No. I think I seen one once. Never got to know her, though. I guess there is good dames, though. Ever played baseball when you was a kid? No, not much. There wasn't any room on the street. Did you? Sure. I played a swell game, too. The other kids used to call me Home Run Orlando. Yeah. We had a good bunch of fellas. Sure, I used to play baseball every Saturday. We had a swell team.  Nah. I ain't either. Why are we here today and other guys outside? I don't know. Kid, I guess... I guess we're here because we didn't use our brains. We tried to be too smart. I can't figure it out. Life, you know, is all complicated. It's got me licked. Duke, if you was to live your life all over again, what would you do? You got me there. Let's see. I don't know.  But not long, Duke. Come on, put a move on. You're coming with me. We're putting you in a cell where you'll be nice and comfy right near the death house. Sorry, Duke. Time to take a little walk. Your last walk. Did no reprieve come? Didn't I get no stay? Nope.  So long, Duke. Sorry you're going, pal. Last mile. Mr. District Attorney, rights finished to rest and record. Mr. District Attorney is a dramatization by Phillips H. Lord. All names of characters used in this program are for obvious reasons fictitious. Mr. District Attorney, subsequent to tonight, will come to you once a week in half hour form commencing June 27th, 10 to 10 30 p.m. Eastern Daylight Saving Time. It will not be heard during the intervening week. We present now Fred Waring, radio. In a brief chat with Paul Douglas, he'll tell you about Pleasure Time, which is scheduled to be heard over these stations at 11 o'clock Eastern Daylight Saving Time beginning next Monday. Are you scared? No, yes, I am scared. A little bit. Well, Fred, you've always been noticed for the, noted for the shortest half hour in radio and Pleasure Time's going to be a new departure for you and the Pennsylvanians. But don't you feel that producing a new show every day is going to be a tough assignment? On the contrary, Paul, I'm very pleased with the idea of five 15-minute spots a week for this reason. We've always had twice as many ideas as we've had time in which to present them. And this is the opportunity we've been looking for to produce all the ideas we've been keeping on the waiting shelf. Well, I can understand that, Fred, but what puzzles me is how do you ever expect to work that enormous gang of yours into a 15-minute show? My memory serves me right. There's Donna Day, Stuart Churchill, Paulie MacLennick, and Fern, Two Bees, and the Honey. And there's Paul Trio with Jimmy Atkins and Patsy Garrett and Gordon Goodman and Jane Wilson. And then the Glee Club. Yes, and the Glee Club. Well, that adds up to a lot of entertainment. Well, that's not all, Paul. We plan to introduce a number of new features too, both script and musical. For instance? Well, with a new idea, we hope will be an entertaining way of presenting the baseball results of the day each day. Oh, you told me about that with music? Yes, of course. That features you, Paul. Well, that's a glad you mentioned it. And among the featured instrumentalists will be Eric Sudey. He's a violinist and arranger, and he's about the tops among the English musicians just arrived in this country. Formerly with Ray Noble, I believe. I met him the other afternoon. I've heard of him from Continental Records. But now how about an inkling of these script plans? Well, I won't tell all, but I will tell you one thing. Poli will be heard in a new role, the role of a great lover. Astounded. Poli McClinty. Paul, that astounded is in parenthesis there. Oh, yes. You speak in an astounded tone. All right. Poli McClinty in an astounded tone. That's right. That's right. You mean that really that croaking crooner, that fog-voiced drummer of yours, is going to be a great lover? Yeah. Poli is going off the deep end. The new love is coming into Poli's life, and her name is Inga Nelson. Isn't that the girl with the Swedish dialect? She is. She's been on the air a few times in guest appearances with Fred Allen. But to my mind, she's one of the finest young comedians of the day.  I know. But anyway, Fred, how in the world would the channel of the page? So how in the world are you ever going to get all these featured soloists, musicians, choral groups, and script spots into 15 minutes? It looks as though it's going to be the fullest 15 minutes in radio. Well, I'll confess, Paul, it sort of had me stumped for a time, but no longer. I'm working out what might be best described as a stagger system. A stagger system? Yes. In fact, it's a staggering stagger system. I'm having insomnia these nights trying to figure out schedules, performers, works with the rangers, having conferences with script writers in order to play a material for our program in advance. You'll be a very busy man, Fred. Yes, I will. A very busy man. Just one more question. What general effect will you attempt to achieve in producing Pleasure Time? Well, the title pretty well describes it, Paul. Just a pleasant 15 minutes with all shades of variety and music. Choral arrangements will predominate with the Glee Club featured, of course. I think we have to get off the air, Fred, so let me say lots of success and Pleasure Time, starring Fred Waring with the Pennsylvanians, will start next Monday evening, ladies and gentlemen. So long, Paul. Seven o'clock Eastern Daylight Saving Time. This is the National Broadcasting Company, RCA Building Radio City, New York.


What's up guys welcome back To the channel now in today's video. I'm going to be giving you a step-by-step plan That's going to help you get clients this week into your program The only way this is going to work is if you actually follow through all of the steps that I lay out here and This is something that I have helped my mastermind members with in the past and Those who follow through with this I've done very well. I've seen someone take this Over the last 12 months And they've generated an extra $50,000 of Revenue to the business. All right, and it's one message Okay, it's going to sound really simple, but again, it's step-by-step. So first what I want you to do So I want you to take a pen and piece of paper All right, I got mine right here Got my journal got my pen. So you need yours too What I want you to do is I want you to write down Right a couple things here. You're gonna write down a nine word email All right, and this email can be used on text email Facebook message Instagram message wherever you do your messaging It doesn't matter to me if you use like software or an app Doesn't matter to me But we need to have this set up to where it gets sent to old prospects old clients Anyone who's on the fence right now Anyone who you are in contact with on social media and I'm talking about parents This is not a message to send kids All right, you guys who are messaging kids all the time and you're not getting clients There's a reason kiddos All right, don't have credit cards. Let's stop selling to kids Let's make that final no more selling the kids does not work. All right, so this message is going out to parents Here is the message. I want you to write this down. All right Is your child still interested With Improving in and it's only blank of the sport that you coach or the problem that you solve Whatever it is Okay, you send that message via an email You send that message via text you send that message via Facebook messenger Instagram DM again, wherever you are communicating with prospects or old clients That message will re-engage those who have been interested To get on a call with you All right, because most people will respond to that email or to that message. They'll say yes Or they will say no they say no move on the next person Okay, but when they say yes, it's not okay. Great. Let's stay on this email thread and go back and forth for the next 19 months No, we need to move them on to a phone call. So that's where we schedule the call with them and Then from there, that's where we sell them the program right now, it's a very simple way of re-engaging old clients Who want your help who need your help and maybe they just haven't heard from you in a while? Or maybe you haven't been direct with them on Getting them back in your program Okay, whatever the case is this message works very Very well, and that's why I wanted to give it to you because I know if you use it It's going to work really well for you. Okay. Here's the thing though This is where a lot of people mess up They will take what I'm talking about and be like, all right I'm just gonna send one email and I'm gonna wait for people to get back to me. All right It doesn't work that way. Okay. The email goes out. Yes, you'll get responses Right. If you only have one person you're emailing All right, that's probably not good enough. We need to start gathering together an email list of contacts People in your area who know I can trust you that those types of people should be getting your message right so we can get a lot of eyeballs on your message and If you just send that email though, that's not going to be the thing that sells in your program It's all going to come down to how you transition that message to get somebody on a phone call That is where the magic is When you get them on a call, what are you saying? Right? What do you actually what are you saying? What are you offering? Are you clear with your offer? Are you clear with your sales? Do you know what you are selling? Yes or no All right. Is this something that is it that you are Giving the the person to talk to is there a sense of urgency? Do they need to get it before a certain date? Are you giving them a discount? How is it set up now that is up to you? Obviously I can help you with that sort of stuff But I need to know more about your business So the only way I can know more about your business is if you shoot me a text at two one zero nine six zero five seven seven one Now if you have a big email list this message here will This will absolutely crush because You're gonna get a lot of people to respond and then you can based off what they say to me help with You can send them digital products. You can send them Private training group training like you can reengage a lot of people back in to your program by sending this one Simple message and this one simple message could be done on social media. You could go on your camera You pull this out. It's free.  We're gonna get people who are interested This these are the sort of things that work All right. Notice how I didn't say anything about creating a logo. I didn't say anything about yeah I didn't say anything about a website this is Direct messaging. All right, we are getting our message in front of the right people at the right time So they respond All right, it's all it is Okay, so I hope this helps you out take what I'm talking about today and use it I promise you this is gonna work. All right. I did the exact same thing. I'm talking about the thing that I'm teaching you here I did that a couple weeks ago. All right. That got four new clients in my program All right, I do that same sort of message at least once a month All right, so and you should do you should have an alarm that goes off and it says nine word message All right, that should go up once a month You should be getting that in front of people who have trained with you in the past or people that are happy on the fence and Anyone on social media who follows you that is something that absolutely works. Go do it You'll thank me later and that's it for today's video guys Go go get that done Shoot me a text. Let me know how it goes if you need any help with like the closing side of things it may be you're good at Getting these phone calls, but you struggle on over the phone If you need help with that, shoot me a text at<PHONE_NUMBER>. I'd love to talk to you. See you later