Your questions answered #2

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Search engines are imperfect. A portion of my visitors ask a specific question, but I am answering a different one. Google isn’t yet smart enough to see the subtleties, and directs them to I hope they found what I wrote interesting, but I admit they must remain unsatisfied.

This is the second time I will be collecting questions that need answers from my search logs. The first time, after a series of posts about math shortcuts, I wanted to help someone with a misconception. This time is two trick questions.

Question: What is one number that’s divisible by everything?

Answer: That is a trick question. Instead of trying to solve for such a number, let me tell you that the number is zero. Through a little thinking, you can agree that it is true. 1/4th minus 1/4th equals zero fourths. That is 0/4ths. No forths at all. Nada. Zero. Zero divided by 4 is zero. 0/2=0, 0/9299292929=0, zero divided by everything is zero. The only exception, is 0/0 = undefined. So with that exception, zero is the one number that is divisible by everything.


Answer: Well since we are looking for any number that has this property, and not just the smallest, we can simply multiply all nine numbers together. 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 = 362880. That is one answer. If you want the smallest answer, then let us try harder. If we are going to multiply by 9, then we don’t need to multiply by 3, since 9 already has a factor of 3. 8 has factors of 4 and 2, so we don’t need to include them either. 7 is prime (no factors other than itself and 1), so we keep that. 6 has factors of 2 and 3. We took care of both factors already since 9 and 8 already had those factors. But that also means the 3 factor for 9 and the 2 factor for 8 will give us divisibility by 6. So we don’t need to multiply by 6. 5 is prime too, and not a factor of any larger number we already included, so we need it. And we already got rid of 4, 3, and 2. 1 doesn’t change anything, so keep or skip it… 9 x 8 x 7 x 5 = 2520, which is then the smallest answer. But wait, is it? The question above already showed us that zero is divisible by everything! So the smallest answer to this question is zer0. Another trick question.

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