Show gcd(a,b) and gcd(a,c) are relatively prime

Question

Let b and c ∈ Z. Suppose that b and c are relatively prime. Show that for all integers a, gcd(a, b) and gcd(a, c) are relatively prime.

Answer 1

For contradiction, assume that exists an integer $a$ such that $\gcd(a,b),\gcd(a,c)$ has a common divisor $d\ge 2$. Then $d\mid \gcd(a,b)\implies d\mid a,b$ and $d\mid \gcd(a,c)\implies d\mid a,c$. But then $d\mid b,c$, so $\gcd(b,c)\ge d\ge 2$, contradiction.