If we have $mr + ns = 1$, how do we prove that $\operatorname{GCD}(m,n) = 1$?

Question

Prove that greatest common divisor of $m$ and $n$ is $1$ where $m<n$.

Answer 1

A common factor of m and n would be a factor of mr+ns=1.

Answer 2

Hint:

If integer $d>0$ divides both $m,n$

$d$ must divide $mr+ns$ for any integers $r,s$

Answer 3

Let $d = \gcd(m,n).$

Then $d$ is a divisor of $m$ and of $n.$

So $da= m$ and $db=n$ for some integers $a,b.$

Therefore $1=mr+ns = (da)r+(db)s = d(ar+bs).$ So $d$ is a divisor of $1.$