The statement feels very obvious when you think about it, but I’m very stuck trying to prove it.
I first tried to prove it by contradiction but I didn’t get anywhere by doing that. I’ve been told that the best way to do it is to just use an example.
Surely there must be a better way of proving it mathematically? I just can’t seem to put my finger on it.
Observation: $gcd(a,b)|a$
This is a direct consequence of the fact that the gcd is a divisor of $a$. Therefore if the gcd is $2$, then $2|a$ so $a$ is even. Similarly, $b$ is even.
The reverse isn’t true. $4$ and $8$ are even but their $gcd$ is $4$, not $2$. However, there’s a simple way to patch the theorem to make it an if and only if. Do you see what it is?