If gcd$(a, 4) = 2$ and gcd$(b, 4) =2$, then gcd$(a + b, 4) = 4$

Question

If gcd$(a, 4) = 2$ and gcd$(b, 4) =2$, then gcd$(a + b, 4) = 4$

can someone help me solve this.

Answer 1

Since $\gcd(a,4)=2$, therefore $a=2k$ with $k$ being odd (otherwise the gcd will be $4$). Likewise $b=2m$ where $m$ is odd. Now $a+b=2(k+m)$. But $k+m$ is even. So now can you deduce the result.

Answer 2

$$\gcd(a, 4) = 2 \rightarrow a=4k+2\\\gcd(b, 4) = 2 \rightarrow b=4q+2\\\gcd(a+b, 4) = \gcd((4k+2+4q+2,4)=\gcd(4k+4q+4,4)\\\gcd(4(q+k+1),4)=4 $$

Answer 3

HINT: Given conditions implies that, there exist $n,m\in\mathbb{N}$ such that $$a=4n+2$$ and $$b=4m+2.$$