Prove or disprove: If $a\mid b+ c$ and $a \mid b - c$ and $a$ is odd, then $a \mid b$. Prove or disprove: If $a\mid b + c$ and $a\mid b - c$ and $a$ is odd, then $a\mid b$.
I cannot seem to find a counterexample so I am thinking it might be true, but cannot prove it either.
This is what I have done so far:
Assume $a\mid b + c$ and $a\mid b - c$ and $a$ is odd. Then $b + c = ak_1$ and $b -c = ak_2$ for integers $k_1,k_2$. Then $c = ak_1-b$ and $c=b-ak_2$. So $ak_1-b=b-ak_2$ $\Rightarrow$ $2b = ak_1+ak_2 = (k_1+k_2)a \Rightarrow a\mid 2b$.
Now I am not sure what to do. Any help is appreciated.

We have $b+c=am_1$ and $b-c=am_2$. Adding both gives us that $2b = a(m_1+m_2)$. This means $a \mid 2b$. However, since $a$ is odd, we have $\gcd(a,2) = 1$. Hence, $a \mid b$.
