Help with a proof???

Question

Suppose a and b are integers that divide the integer c. If a and b are relatively prime (meaning gcd(a,b)=1), show that ab divides c.

Thanks in advance for any help you can offer.

Answer 1

List the prime factors of a: p1, p2, p3, etc

List the prime factors of b: q1, q2, q2, etc.

Since g c d (a,b) = 1, the lists are mutually exclusive (none in common).

But a and b divide c.

Hence all the p's and all the q's are factors of c.

Therefore ab = (product of all of them) divides c.