2.1 Topics in Number Theory

2.1 Topics in Number Theory

2.1.1 Divisibility

The set of integers {... ,-3, -2, -1, 0, 1, 2, 3,...} is denoted by symbol Z.

Definition 2.1.1 Let a, b be integers. Then a divides b (equivalently: a is a divisor of b,

or a is a factor of b) if there exists an integer c such that b = ac.ifa divides b, then this

is denoted by a|b.

Example 2.1.1 (i) -3|18, since 18 = (-3)(-6). (ii) 173|0, since 0 = (173)(0).

The following are some elementary properties of divisibility.

Proposition 2.1.1 (properties of divisibility)

1. a|a.

2. If a|b and b|c, then a|c.

3. If a|b and a|c then a|(bx + cy) for all x, y Z.

4. If a|b and b|a, then a = ±b.