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.