Problem regarding GCD

Question

Let $a, b, m$ and $n$ be integers, prove that if $a \mid m$ and $b \mid n$ then $\text{gcd}(a,b) \mid \text{gcd}(m,n)$.

How can prove that a gcd could divide another gcd?

Answer 1

Sine $a \mid m \implies m = ka$, and also since $b \mid n \implies n = pb$. Now let $x = \text{gcd}(a,b)$. Then $x \mid a \implies x \mid m$. Similarly, $x \mid n$. Thus $x \mid \text{gcd}(m,n)$, or $\text{gcd}(a,b) \mid \text{gcd}(m,n)$.