So I am currently trying to prove some basic divisibility relations, as follows.
- If $a \mid b$ and $a \mid c$, then $a \mid b + c$.
- If $a \mid b$ and $s \in \mathbb{Z}$, then $a \mid sb$.
- If $a \mid b$ and $a \mid c$ and $s$, $t \in \mathbb{Z}$, then $a \mid sb + tc$.
- If $a \mid b$ and $b \mid c$, then $a \mid c$.
- $a \mid 0$ for all $a \neq 0$.
- $1 \mid b$ for all $b \in \mathbb{Z}$.
- If $a \mid b$ and $b \neq 0$, then $|a| \le |b|$.
- If $a \mid b$, then $\pm a \mid \pm b$.
I frequently find myself having trouble showing these quite basic facts.
- What should I keep in mind when trying to prove these properties, i.e. what techniques are useful?
- What is the intuition for the proofs of these facts, or rather, morally why must these facts be true?
Thanks in advance.
Good question.
It may be easiest to prove some of these facts straight from the definition. That is, recall that $a|b\implies \exists k\in\mathbb{Z}$ such that $ak=b$. For your first property, we have $ak=b$ and $al=c$ for some $k,l\in\mathbb{Z}$.
When we add those two together we find $ak+al=b+c$. Then, by distributivity, $(k+l)a=b+c$. Since $k+l\in\mathbb{Z}$, we have that $a|(b+c)$ by definition.
Hopefully that provides some framework to prove some of these other statements since many amount to simple algebraic manipulation once you apply the definition of "divides."