Question

4. Let a, b, c be integers.

(a) Prove if gcd(ab, c) = 1, then gcd(a, c) = 1 and gcd(b, c) = 1. (Hint: use the GCD characterization theorem.)

(b) Prove if gcd(a, c) = 1 and gcd(b, c) = 1, then gcd(ab, c) = 1. (Hint: you can use the GCD characterization theorem again but you may need to multiply equations.)

(c) You have now proved that “gcd(a, c) = 1 and gcd(b, c) = 1 if and only if gcd(ab, c) = 1.” Is this statement true if the greatest common divisor is not 1?

So is the following statement true: “gcd(a, c) = d and gcd(b, c) = d if and only if gcd(ab, c) = d?”

Answer 1