if $(a,4)=(b,4)=2$, then $(a+b,4)= 4$

Question

Simple question and I did it too by brute force but is there way to do using identities. Question is if $(a,4)=(b,4)=2$, then $(a+b,4)= 4$, where $(a,b)$ stands for the gcd of $a$ and $b$. Clearly possibilities of $a$ and $b$ are {2,6}, so it holds, but as I said, if values were higher how to approach using identities. Thanks.

Answer 1

Because $(a,4) = (b,4) = 2$, we have $a = 2m$ and $b = 2n$, where $m$ and $n$ are odd numbers and $m + n = 2k$ is even.

Then, $(a+b,4)=(2(m+n),4) = (4k,4) = 4$