Prime Factorization of Whole Numbers

Examples showing how to find the prime factors of a positive integer are presented below. You may also use the prime factorization calculator to check your answers. Practice questions and complete solutions are included.

Factors

The factors of a positive integer \( n \) are all positive integers that divide \( n \) with remainder zero.

For example, \( 2, 3, 4 \), and \( 6 \) are factors of \( 12 \) because:

\[ \begin{aligned} 12 \div 2 &= 6 \\ 12 \div 3 &= 4 \\ 12 \div 4 &= 3 \\ 12 \div 6 &= 2 \end{aligned} \]

These factors also appear when \( 12 \) is written as a product:

\[ 12 = 6 \times 2 = 4 \times 3 \]

Prime Factorization

The Fundamental Theorem of Arithmetic states that every positive integer greater than 1 can be written in exactly one way as a product of prime numbers.

A prime number is a positive integer greater than 1 that has exactly two positive divisors: \( 1 \) and itself.

The first few prime numbers are:

\[ 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, \ldots \]

How to find prime factors:
Start dividing by \( 2 \) as long as possible, then continue with the next prime numbers \( 3, 5, 7, \ldots \) until the remaining quotient is prime.

Examples

Example 1

Factor \( 4 \) into prime numbers.

\[ 4 = 2 \times 2 = 2^2 \]

Example 2

Factor \( 20 \) into prime numbers.

\[ \begin{aligned} 20 &= 2 \times 10 \\ &= 2 \times 2 \times 5 \\ &= 2^2 \times 5 \end{aligned} \]

Example 3

Factor \( 100 \) into prime numbers.

\[ \begin{aligned} 100 &= 2 \times 50 \\ &= 2 \times 2 \times 25 \\ &= 2^2 \times 5^2 \end{aligned} \]

Example 4

Factor \( 1020 \) into prime numbers.

\[ \begin{aligned} 1020 &= 2 \times 510 \\ &= 2^2 \times 255 \\ &= 2^2 \times 3 \times 85 \\ &= 2^2 \times 3 \times 5 \times 17 \end{aligned} \]

Example 5

Factor \( 634 \) into prime numbers.

\[ 634 = 2 \times 317 \]

Example 6

Factor \( 720 \) into prime numbers.

\[ \begin{aligned} 720 &= 2 \times 360 \\ &= 2^2 \times 180 \\ &= 2^3 \times 90 \\ &= 2^4 \times 45 \\ &= 2^4 \times 3^2 \times 5 \end{aligned} \]

Practice Questions

Part A

Which of the following is not a prime factorization?

\[ \begin{aligned} \text{a)}&\; 2 \times 2 \times 4 \quad \text{b)}\; 3 \times 3 \times 5 \times 9 \\ \text{c)}&\; 3 \times 3 \times 5 \times 17 \quad \text{d)}\; 2 \times 5 \times 5 \times 21 \\ \text{e)}&\; 2 \times 2 \times 3 \times 5 \times 41 \end{aligned} \]

Part B

Factor each number into prime numbers:

\[ 18,\; 300,\; 123,\; 1200,\; 1450 \]

Solutions

Part A

  1. \( \quad 4 \) is not a prime number , therefore \( 2 \times 2 \times 4 \) is not a prime factorization
  2. \( \quad 9 \) is not a prime number , therefore \( 3 \times 3 \times 5 \times 9 \) is not a prime factorization
  3. \( \quad 3 \times 3 \times 5 \times 17 \) is a prime factorization because \( 3, 5 \) and \( 17 \) are prime numbers
  4. \( \quad 21 \) is not a prime number , therefore \( 2 \times 5 \times 5 \times 21 \) is not a prime factorization
  5. \( \quad 2 \times 2 \times 3 \times 5 \times 41 \) is a prime factorization because \( 2, 3, 5 \) and \( 41 \) are prime numbers.

Part B

\[ \begin{aligned} 18 &= 2 \times 3^2 \\ 300 &= 2^2 \times 3 \times 5^2 \\ 123 &= 3 \times 41 \\ 1200 &= 2^4 \times 3 \times 5^2 \\ 1450 &= 2 \times 5^2 \times 29 \end{aligned} \]

More References