If $\gcd(a,4)=\gcd(b,4)=2$, find $\gcd(a+b,4)$.

Question

If the greatest common divisor (GCD) of $a$ and $4$ is $2$, and that of $b$ and $4$ is $2$, what is the GCD of $a+b$ and $4$?

I tried writing $4$ as $2^2$. So GCD of $a$ and $2^2$ is $2$ and GCD of $b$ and $2^2$ is $2$. So the GCD of $a+b$ and $2^2$ is also $2$.

Answer 1

$(a,4)=2\iff a=2a_1$ with $a_1$ odd.

$(b,4)=2\iff b=2b_1$ with $b_1$ odd.

$(a+b,4)=(2(a_1+b_1),4)=4$, since $a_1+b_1$ is even.