For the first part, I got this from Niven 5th edition:
There exist integers such that
Thus,
and gives....
Rearranging, we get:
which is in the form
and in the form of the gcd.
Thus
Let n and m be coprime positive integers. Show that if n divides a and m divides a then nm divides a. Deduce that nm is the least common multiple of n and m.
I have shown the first part by expressing a as a product of distinct prime numbers but not the deduction.
I'm also having trouble with the following;
Suppose that n and m are arbitrary positive integers. Show that if n divides a and m divides a then nm/(n,m) divides a. Deduce that nm/(n,m) is the least common multiple of n and m.
Thank-you for your help.
For the second part, this may help:
Theorem:
Proof:
and thus we can assume
Let
Thus and ...which gives... and for some integers
Let
then claim
here we have shown that is a common multiple of and .
Let be some common multiple of and .
integers such that and
Since then
but
Thus which is an integer.
This proves that any generic common multiple is divisible by and we can conclude that is the
Thus