Euclide arithmetic, divisibility

Question

If $a_1,\dots,a_n$ are non zero integers, prove that if $d$ divides $a_i$, for $i=1,\dots,n$, then $d$ divides $a_1+\dots+a_n$.

I can see that if it divides each term individually it will divide the sum, but I don't know how to prove it.

Answer 1

Here is a fundamental and generalization rule of your thoughts:

If $c$ divides $a$ and $b$, then it divides any linear combination $ax+by$ for any integers $x,y$.

The proof of this statement goes as follows:

from the definition of divisibility $a = ck$ for some integer $k$, implying that $ax = cK, K = xk$. Similarly, $by = cR$ for some integer $R$. Adding these two equations, $ax+by = c(R+K)$. We know that $(R+K)$ is an integer, let us write it $P$. Thus $ax+by = cP$ implying that $c$ divides $a+b$.

Now you can use this fact to prove your statement with $x,y = 1$, can't you?