Thread: Suppose that gcd(a,b)=1 and that a|n and b|n. Prove that ab|n.

Suppose that gcd(a,b)=1 and that a|n and b|n. Prove that ab|n.

Originally Posted by qtpipi

Suppose that gcd(a,b)=1 and that a|n and b|n. Prove that ab|n.

If gcd(a,b)=1 then there are integers r,s such that ra + sb = 1. If a|n and b|n then there are integers k,l such that n = ka = lb. You can then deduce that
rka + skb = k,
b(rl + sk) = k,
ab(rl + sk) = ka = n,
so n is a multiple of ab.