Suppose $a$ and $b$ are nonzero and both even. Prove or disprove the following: $\gcd(a,b)$ is even.

Question

Proof: Suppose $a$ and $b$ are nonzero and both even. Prove or disprove the following: $\gcd(a,b)$ is even.

Can someone give me some insight into this proof? I've done plenty of examples and I believe this to be true, but when proving it I get stumped.

What I know:

d | a <=>  a = dk, for some integer k

d | b <=> b = dk, for some integer k

Answer 1

$d=\gcd(a,b)$ is a number such that $d\mid a$ and $d \mid b$. Further, if $x\mid a$ and $x\mid b$ then $x\mid d$.

$2\mid a$ and $2\mid b$. Conclude.

Answer 2

Hint $ $ An odd common divisor $d$ can't be greatest since $\,2d$ is also a common divisor (by $\,2\mid a,b)$

Answer 3

Well, if we write $\gcd(a,b)=d$, then Bezout's Theorem tells us that $$ax+by=d$$ for some integers $x$ and $y$. Because $a$ and $b$ are both even, it is easy to see that $ax+by$ is also even. This implies that $d$ must be even.