The integers
n
and
q
are called
factors
of
m
.
If the division does not come out even, the remainder is less than the number
we tried to divide. For example, 16 divided by 5 is 3 with a remainder of 1.
This whole collection of elementary school flashbacks can be summarized in
a statement that sounds far more impressive than “long division,” namely, the
Division Algorithm
.
The Division Algorithm.
Suppose n and m are natural numbers. Then there exist unique numbers
q (for quotient) and r (for remainder), that are either natural numbers
or zero, such that
m
5
nq
1
r
and
0
r
n
2
1 (r is greater than or equal to 0 but less than or equal to n
2
1).
Prime Time
Factoring a big number into smaller ones gives us some insights into the larger
number. This method of breaking up natural numbers into their basic compo-
nents leads us to the important notion of prime numbers.
There certainly are natural numbers that cannot be factored as the product of
two smaller natural numbers. For example, 7 cannot be factored into two smaller
natural numbers, nor can these: 2, 3, 5, 11, 13, 17, which, including 7, are the first
seven such numbers (let’s ignore the number 1). These unfactorable numbers are

Number Contemplation
70
called
prime numbers
. A prime number is a natural number greater than 1 that
cannot be expressed as the product of two smaller natural numbers.
The prime numbers form the multiplicative building blocks for all natural
numbers greater than 1. That is, every natural number greater than 1 is either
prime or it can be expressed as a product of prime numbers.
The Prime Factorization of Natural Numbers.
Every natural number greater than 1 is either a prime number or it can be
expressed as a product of prime numbers.
Let’s first look at a specific example and then see why the Prime Factorization
of Natural Numbers is true for all natural numbers greater than 1.
Is 1386 prime? No, 1386 can be factored as
1386
2
693
H11005
H11003
.
Is 693 prime? No. It can be factored as
693
3
231
H11005
H11003
.
The number 3 is prime, but
231
3
77
H11005
H11003
.
So far we have
1386
2
3
3
77
H11005
H11003
H11003
H11003
.
Is 77 prime? No,
77
7
11
H11005
H11003
but 7 and 11 are both primes. Thus we have
1386
2
3
3
7
11
H11005
H11003
H11003
H11003
H11003
.
So, we have factored 1386 into a product of primes. This simple “divide and
conquer” technique works for any number!
Proof of the Prime Factorization of Natural Numbers
Let
n
be any natural number greater than 1 that is not a prime. Then there must be
a factorization of
n
into two smaller natural numbers both greater than 1 (why?),
say
n
5
a
b
. We now look at
a
and
b
. If both are primes, we end our proof, since
n
is equal to a product of two primes. If either
a
or
b
is not prime, then it can be
factored into two smaller natural numbers, each greater than 1. If we continue in
Understanding a specific
case is often a major step
toward discovering
a general principle.

#### You've reached the end of your free preview.

Want to read all 96 pages?