Prove that there exist integers $a$ and $b$, so $am + bn = 1$.

Question

When $m,n\in \mathbb{Z} $ and $a,b \in \mathbb{Z}$, show that there exist integers $a$ and $b$, so $am + bn = 1$ which also means $gcd(m,n) = 1$.

I tried proof through contradiction but I didn't succeed. Any suggestion would be helpful.

Answer 1

$\textbf{Counterexample}$: Let $m = 0$ and $n = 2$. Then you are suggesting that there exists an integer $b$ such that $2b = 1$.

You need to change the statement to say: If $m,n$ are relatively prime, then there exists $a,b \in \mathbb{Z}$ such that $am+bn=1$. (This is actually bi-conditional)