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 ofgcd(a,b)=1 is necessary for this to work?