3.1 Basic Notions

3.1.1 Divisibility

Number theory is concerned with the properties of the integers. One of the most important is divisibility.


Let a and b be integers with a0. We say that a divides b,  if there is an integer k such that b=ak. This is denoted by a|b. Another way to express this is that b is a multiple of a.


3|15,  15|60,  718 (does not divide).

The following properties of divisibility are useful.


Let a, b, c represent integers.

  1. For every a0,  a|0 and a|a. Also, 1|b for every b.

  2. If a|b and b|c,  then a|c.

  3. If a|b and a|c,  then a|(sb+tc) for all integers s and t.

Proof. Since 0=a0,  we may take k=0 in the definition to obtain a|0. Since a=a1,  we take k=1 to prove a|a. Since b=1b,

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