Proving with divisibility

Question

I have never written any proofs (except high school geometry) in my life, so I'm not sure what exactly the proper formatting should be.

Involving divisibility, the proposition states:

Let $a, b,$ and $c$ be integers. If $a|b$ and $a|c$ then $a|(b+c)$

This is what I wrote, but I'm really not sure if I'm going the right direction with this.

Let $m$ and $n$ be integers

Let $b = am$ and $c = an$

$b + c = am + an = a (m + n)$

Let integer $k = m + n$

$b + c = a (k)$

Then, $a | (b + c)$

I'm starting a proving class this semester without any background or experience in proving. Any advice would be appreciated.

Answer 1

Let m and n be integers
Let b = am and c = an

Here I can see you're using the definition of divisibility, but I'd write it differently: "Suppose $a,b,c$ are integers such that $a | b$ and $a | c$. Then $b=am$ and $c=an$ for some integers $m,n$."

The important difference is that the hypothesis of the problem tells you to start with $a,b$ and $c$, and then the conditions on $a,b,c$ determine $m$ and $n$, not the other way around.

b + c = am + an = a (m + n)

This line looks good, and it's the key step on the proof.

Let integer k = m + n
b + c = a (k)

This is right, but I don't think it's necessary; you can actually end at your previous line because since $m$ and $n$ are integers, you know that $m+n$ is an integer too, so you have written $b+c$ as $a$ times an integer already, which means, as you conclude, that $a$ divides $b+c$.