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## Division

Definition 6.2.14   Given two integers , , we say that divides (or that is a divisor of ) and write if and there exists an integer such that . The number is called a proper divisor of if is not or .

For example, is a proper divisor of and 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 and are primes, and the number is not a prime as . More on primes will be seen further (v.i. section 5.

Proposition 6.2.16   The relation is a partial ordering in .

Proof. The proof that is an ordering goes exactly as in Chapter 4, Example 4.3. As , we see that this ordering is partial.

Proposition 6.2.17   Let , and be three integers. If and then for any two integers and , we have: .

Proof.

Then for any two integers and , we have:

i.e. .

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root 2002-06-10