If $(a,4)=2=(b,4)$, prove $(a+b,4)=4$. I'm almost embarrassed to be asking about a problem such as this one (exercise 12 in Niven 1.2), but here goes:

Given $(a,4)=(b,4)=2$, show that $(a+b,4)=4$.

I have plenty of tricks for working with gcds multiplicatively, however I have honestly no idea how to attack this problem. I sort of halfheartedly tried using Bezout's identity, writing $ax_0+4y_0 = 2$ and $bx_1+4y_1 = 2$. Adding the equations gives
$$ax_0+bx_1+4(y_0+y_1) = 4$$
and hence if I were able to use that $x_0=x_1$ I'd only have to show minimality. However, seeing the exercises around this one I am completely sure I am overcomplicating things here. What am I missing?

$2$ divides both of $a$ and $b$, but $4$ doesn't, then $a,b$ are congruent with $2$ (mod $4$), then $a+b$ is a multiple of $4$. 
