Problem - 3864
(Bezout's theorem) Show that two positive integers $a$ and $b$ are co-prime if there exist integer $x$ and $y$ satisfying $ax+by=1$.
If this is not true, let $d > 1$ be a divisor of $a$ and $b$. This means the left side of $ax+by=1$ is a multiple of $d$. However, the right side cannot be. This is a contradiction which means $a$ and $b$ have to be relatively prime.