Why is it the case that the common factors of $x$, $y$ are also common factors of $x + y$?
For example, $10$, $4$ share factors $2$, $1$. $10 + 4 = 14$ which has factors $2$, $1$
Similarly, $100$, $160$ share factors $1$, $10$, $2$, $4$, and $260$ has factors $1$, $10$, $2$, $4$.
If $d$ divides both $x$ and $y$, then $d$ divides $x+y$ as well. This is true because if $x = da$ and $y=db$ for integers $a,b$, then $x+y=d(a+b)$.