Finding the Prime Factorization of a Number
In algebra, we often need to factor expressions to solve equations or investigate the behavior of functions. An introductory form of this factorization is called prime factorization of whole numbers. A prime number is a whole number with exactly two distinct positive integer factors. These factors are precisely the number itself and 1.
Sample prime numbers:
2 (the only even prime), 3, 5, 7, 11, 13, 17, 19, 23, and 31
Notice that one of the factors is always 1 and the other factor is the whole number itself.
NOTE: 1 is not a prime number because it has only one distinct positive integer factor, namely, 1.
A composite number is a whole number with more than two distinct positive integer factors.
Samples of composite numbers:
a) 4 has factors: 1, 2, 4
b) 12 has factors: 1, 2, 3, 4, 6, 12
c) 15 has factors: 1, 3, 5, 15
d) 51 has factors: 1, 3, 17, 51
e) 60 has factors: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
We can write every composite number as a product of prime numbers. This is called the prime factorization of a number.
Examples
Find the prime factorization of the composite in samples a – e above.
Solutions
A factor tree can help simplify the prime factorization process:
a) 4
b) 12
12
2 6
2 3
c) 
d) 
e) 60
60
2 30
2 15
3 5
