# How do I show that $a \mid b$ and $a \mid c$ implies that $a \mid (b+c)$?

I'm not even sure how this property is called. I read about it in a couple of places, but I can't find the canonical name for it. Here it goes:

If $$a \mid b$$ and $$a \mid c$$ then $$a \mid (b+c)$$, where $$\mid$$ means divides.

I'm wondering why it is true. The similar property with multiplication makes sense for me because of the fundamental theorem of arithmetic, I think. But we're dealing with addition here, so my first reaction is what if the resulting number is no longer odd/even? I'm not sure how to formulate the proper question here. Don't we mess up the underlying (prime) factors of numbers when we add/subtract them?

I was able to mentally connect this with the fraction's addition property, but I'm still not sure why that works.

If possible could you please not try to explain this via complex proofs? I'm a beginner and this will confuse me even more.

If $$a |b$$ and $$a|c$$, there are integers $$k,l \in \mathbb{Z}$$ with $$ak = b$$ and $$al = c$$. Then $$b+c = ak + al = a(k+l)$$ so we see that $$a|(b+c)$$.

Simple, right?

• Thanks! Looks simple, but I'm still gonna need some time to process it😅. So is there a name for this property? May 10 '20 at 14:46
• This is the definition of divisibility? $a |b \iff \exists k \in \mathbb{Z}: ak = b$ May 10 '20 at 14:46
• Think about in the following way: $a|b$ means that $\frac{b}{a} \in \mathbb{Z}$, so there is an integer $k \in \mathbb{Z}$ with $\frac{b}{a}= k \iff b = ka$. May 10 '20 at 14:48
• I think OP is referring to the property in the question, where if two numbers are divisible by the same number then so is their sum. May 10 '20 at 14:54
• I don't think that has a name. The statement follows so easily from the definition that I also think it doesn't deserve a name. Most people would use it as something trivial. May 10 '20 at 14:56

Imagine you add two numbers that are both divisible by the same number, for example $$20$$, $$30$$, and $$5$$.

$$20$$ is divisible by $$5$$. So is $$30$$. So if you add $$20$$ and $$30$$, do you think it will be divisible by $$5$$?

It's true that adding numbers has the potential to ruin their prime factorization, but that becomes less of an issue when they have something in common. For example, $$100 + 1$$ has a completely different factorization than $$100$$ and $$1$$ ($$101$$ is prime!); but not much changes when you consider $$100 + 100$$ instead.

All this is saying is that things that have something in common (usually) keep that thing in common when you combine them.

• Interesting, thanks! There are a lot of stuff written about primes and multiplication, but I couldn't find any similar info about addition/subtraction. Do you know where I could read more about the thing you're talking about, specifically "things that have something in common (usually) keep that thing in common when you combine them"? May 10 '20 at 15:07
• I don't know if someone has formalized a general principle about this. I was just trying to note that this is a property that the divisibility relation has. I'm inclined to say that it's not true for all properties (although I don't have an immediate example)—it just happens to have a very clear interpretation in this case. May 10 '20 at 15:16
• thank you, I appreciate the explanation! May 10 '20 at 19:05