$am+bn = 36$; What can you say about $gcd(m, n)$

Question

Let $a, b, m,$ and $n$ be integers and suppose that $am+bn = 36$. What can you say about $gcd(m, n)$?

The professor said the question has one answer and he gave me this hint.

HINT: $gcd(m, n)$ has the property that it divides both $m$ and $n$. What does imply about $gcd(m, n)$ and $am+bn$?

My answers: $gcd(a, b) = gcd(m, n)$. And $gcd(m, n)$ and $gcd(a, b)$ are both divisible by $36$.

Uhm... I'm not sure what hint implies and if my answer is correct..

Answer 1

The answer is $$\gcd(m,n)|36$$ This is because $am+bn$ is divisible by $\gcd(m,n)$.

Answer 2

No. It's the other way around. Since $gcd(m,n)$ divides both $m$ and $n$, it divides $36$.

That's about all that can be said...