Divisors of Product of Coprime Integers

From ProofWiki

Contents 1 Theorem 1.1 Corollary 2 Proof 2.1 Proof of Corollary

[edit] Theorem

Let , where .

Then , where and .

[edit] Corollary

Let be a prime.

Let , where .

Then or , but not both.

[edit] Proof

Let .

By Divide by GCD for Coprime Integers, .

So we have written where divides .

We now show that divides .

Since divides there exists such that .

Substituting for and :

which gives:

So divides .

But so by Euclid's Lemma as required.

[edit] Proof of Corollary

From the main result, , where and .

But as is prime, either:

and , or:

and .

So or .

But can not divide both as .