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

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

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

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


next up previous contents
Next: The Euclidean Division Up: The integers Previous: Ordering   Contents
root 2002-06-10