If a l gcd(b,c), then gcd(ab,c) = gcd (b,c)

the question i wanna ask is
1. if a l gcd(b,c), then gcd(ab,c) = gcd (b,c)
Is it true or false? If true, please give me a proof, and if false, please explain why it is false.

Thanks!

If a l gcd(b,c) = u say, the b = mu and c = nu where (m,n) = 1,so a = ku, for some k.
so ab = km(u^2) and c = nu
gcd(ab,c) = gcd( km(u^2), nu) = pu where p = gcd(k,n) so it is true.