Part I. Integers:
(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
.
(Law of Identity Elements)
For all integer
a
,
a
+ 0 = 0 +
a
=
a
;
a
• 1 = 1 •
a
=
a
.
(Law of 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.
Theorem.
The sum of two odd integers is an even integer.
Theorem.
If
a

b
, and
b

c
, then
a

c
, where
a
,
b
, and
c
are integers.
Definition (Prime Number)
A positive integer
n
> 1 is a prime number if its divisor must be either 1 or
n
itself (that is, if
m

n
, then
m
= 1 or
m
=
n
).
Theorem (Fundamental Theorem of Arithmetic)
Any integer
n
> 1 can be written as a product of prime numbers.
Further, this product is unique except for rearrangement of the terms.
Theorem (Euclid’s Division Theorem)
For any integers
n
and
m
,
m
≥
1, there exist integers
q
and
r
, 0
≤
r
<
m
, such that
n
=
m q
+
r
.
Further, these integers
q
and
r
are unique.
Theorem.
Every integer
n
is either even or odd.
Theorem.
If
a
,
b
are two integers and
a b
is even, then at least one of
a
and
b
is even.
Theorem
(The Pigeonhole Principle)
If
n
+ 1 or more pigeons are nested in
n
holes,
n
≥
1, then at least one hole nests
more than one pigeon.
Theorem.
The number
√
2 (the square root of 2) is not a fraction of two integers (i.e., not a
rational
number).
Theorem.