First, let us recall some related definitions and theorems in the domain of integers.
The set of integers,
Z
, contains numbers 0,
–
1, +1,
–
2, +2, -3, +3, etc.
That is,
Z
= { …,
–
2,
–
1,
0, 1, 2, … }.
The set of integers and the three operations + (add), – (subtract), and • (multiply),
satisfy the following properties:
(Closure Property under +, –, •)
If
a
,
b
are integers, then
a
+
b, a
–
b, a
•
b
(or simply
ab
) are
integers.
(Commutative Law)
If
a
,
b
are integers, then
a
+
b = b
+
a
,
a
•
b
=
b
•
a
.
(Associative Law)
If
a
, b, c are integers, then (
a
+
b
) +
c
=
a
+ (
b
+
c
), and (
a
b
)
c
=
a
(
b
c
).
(Distributive Law)
If
a
, b, c are integers, then
a
(
b
+
c
) =
a b
+
a c
.
(Identity Elements)
For all integer
a
,
a
+ 0 = 0 +
a
=
a
;
a
• 1 = 1 •
a
=
a
.
(Additive Inverse)
For all integer
a
,
a
+ (–
a
) = (–
a
) +
a
= 0.
Definition (Divisibility)
If
a
,
b
are integers, then
a | b
, or
a
is a divisor of
b
, if
a
≠
0
and there
exists an integer
c
such that
b
=
a c.
Definition (Even and odd)
An integer
a
is even if there exists an integer
b
such that
a
= 2
b
(or, equivalently, if 2 |
a
); an integer
a
is odd if there exists an integer
b
such that
a
= 2
b
+ 1.