Dear forum members,  I am currently doing a maths project and as
a part of this project I am trying to prove a certain conjecture
about the divisibility of numbers.  First of all, could someone
please take a look at the proof given in this document on page
4, solution 2.  I do not understand how does Euclid's lemma
relate to the proof given, and I'd be grateful if someone could
explain this to me.  Second, if I was trying to prove a certain
thing about the divisibility of two integers, call them a and b,
by a number x, and if I knew that the difference of these
integers, a-b, is divisible by x, could I use the congruence
theorem of modular arithmetic to prove that both a and b are
divisible by x?  I am really lost in the world of proofs and do
not understand them at all. So if someone could please help me,
I would be very grateful.  Thank you!