Congruence

Consider the integers Z := {. . . ,−2,−1, 0, 1, 2, . . .}. For a, b 2 Z, we say
that b divides a, or alternatively, that a is divisible by b, if there exists
c 2 Z such that a = bc. If b divides a, then b is called a divisor of a, and
we write b | a. If b does not divide a, then we write b - a.
We first state some simple facts:
Theorem 1.1. For all a, b, c 2 Z, we have
(i) a | a, 1 | a, and a | 0;
(ii) 0 | a if and only if a = 0;
(iii) a | b and a | c implies a | (b + c);
(iv) a | b implies a | −b;
(v) a | b and b | c implies a | c.