Prove that if gcd(a,b) = 1 and gcd(a,c) = 1 then gcd(b,c) = 1

I assume you use the fact that since gcd(a,b) = 1 then 1 = as + bt for some integers s,t
and since gcd(a,c) = 1 then 1 = aq + cr for some integers q,r
And then using aq + cr = as + bt or bt - cq = a (r - s) we can figure out that gcd(b,c) = 1 but I'm not sure if I am on the right track or not.

The statement is not actually true. For example

a=2, b=5, c=15

gcd( 2, 5 ) = 1, and gcd( 2, 15 ) = 1, but gcd( 5, 15 ) = 5