Suppose that gcd(a,b)=1, a|c and b|c. Prove that ab|c.
I've already got this part:
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.
rka + skb = k,
b(rl + sk) = k,
ab(rl + sk) = ka = c,
so ab | c
How can I show that the initial assumption of gcd(a,b)=1 is necessary for this to work?