Let p be prime and suppose that p divides the product abc in

Let p be prime, and suppose that p divides the product abc in Z. Prove that p divides one of a, b, or c.

Solution

Suppose p does not divide b and c

Then p, b and c are relatively prime.

Therefore, gcf(p, b, c) = 1

Since p divides abc and gcf(p, b, c) = 1

So, p divides a.

So we have proved p divides a or p divides b and c.