Thinking
Mathematically
Number Theory: Prime and
Composite Numbers
The Set of Natural Numbers
N = {1,2,3,4,5,6,7, 8, 9, 10, 11, ... }.
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Divisibility
If a and b are natural numbers, a is divisible
by b if the operation of dividing a by b
leaves a remainder of 0. This is the same as
saying that b is a divisor of a, or b divides
a. All three statements are symbolized by
writing b|a.
Prime Numbers
A prime number is a natural number greater
than 1 that has only itself and 1 as factors.
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Composite Numbers
A composite number is a natural number
greater than 1 that is divisible by a number
other than itself and 1.
The Fundamental Theorem of
Arithmetic
Every composite number can be expressed as a
product of prime numbers in one and only one
way (if the order of the factors is disregarded).
For example 700 can be written in the following way.
700 = 2 x 2 x 5 x 5 x 7
The prime factors of 700 are 2, 5, and 7.
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“Factor Trees”
The prime factors of a natural number can be found by
constructing a “factor tree.” Write the given number
as a product and continue to factor each composite
number until only prime numbers remain.
The “prime factorization” of
40
40 is determined by the prime
8 x
5 numbers at the bottom of each
branch of the tree.
40 = 2 x 2 x 2 x 5 = 23 x 5
4 x 2
2
x
2
The prime factors of 40 are 2
and 5.
Finding the Greatest Common
Divisor of Two or More Numbers
Using Prime Factorization
To find the greatest common divisor of two or more
numbers:
1. Write the prime factorization of each number.
2. Select each prime factor with the smallest
exponent that is common to each of the prime
factorizations.
3. Form the product of the numbers from step 2.
The greatest common divisor is the product of
these factors.
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Example of Finding the Greatest
Common Divisor
40
x
x
2
5
6
x
4
3 x 22 x 2
40 = 2 x 2 x 2 x 5
x 2
24 = 2 x 2 x 2 x 3
The greatest common divisor of 24 and 40
is 2 x 2 x 2 = 8.
4
2
8
24
Finding the Least Common Multiple
Using Prime Factorization
To find the least common multiple of two or more
numbers:
1. Write the prime factorization of each number.
2. Select every prime factor that occurs, raised to
the greatest power to which it occurs, in these
factorizations.
3. Form the product of the numbers from step 2.
The least common multiple is the product of
these factors.
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Example of the Least Common
Multiple
40
4
2
8
x
x
2
12
5
3
x
4
2
x
2
40 = 2 x 2 x 2 x 5
x 2
12 = 2 x 2 x 3
The least common multiple of 12 and 40 is
2 x 2 x 2 x 3 x 5 = 120.
Thinking
Mathematically
Number Theory: Prime and
Composite Numbers
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