Division

Definition 6.2.14   Given two integers $a$, $b \in \mathbb{Z}$, we say that $a$ divides $b$ (or that $a$ is a divisor of $b$) and write $a \mid b$ if $a \neq 0$ and there exists an integer $q$ such that $b = a \cdot q$. The number $a$ is called a proper divisor of $b$ if $a$ is not $\pm 1$ or $\pm b$.

For example, $4$ is a proper divisor of $20$ and $5$ has no proper divisor.

Definition 6.2.15   A prime integer or, more briefly a prime is an integer which has no proper divisor.

For example, the numbers $17$ and $131$ are primes, and the number $48$ is not a prime as $12 \mid 48$. More on primes will be seen further (v.i. section 5.

Proposition 6.2.16   The relation $\mid$ is a partial ordering in $\mathbb{Z}$.

Proof. The proof that $\mid$ is an ordering goes exactly as in Chapter 4, Example 4.3. As $3 \not\mid 5$, we see that this ordering is partial. $\qedsymbol$

Proposition 6.2.17   Let $a,$, $b$ and $d$ be three integers. If $d \mid a$ and $d \mid b$ then for any two integers $m$ and $n$, we have: $d \mid ma + nb$.

Proof.

$$\begin{cases} d \mid a \\ d \mid b \end{cases} \Longleftr... ...thbb{Z}\; a=dp \\ \exists q \in \mathbb{Z}\; b=dq \end{cases}$$

Then for any two integers $m$ and $n$, we have:

$$ma+nb=m(dp)+n(dq)=d(mp+nq)$$

i.e. $d \mid ma + nb$. $\qedsymbol$