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G.3 Prime factorization

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What is prime factorization?

Prime factorization is finding which prime numbers multiply together to make a given number. Every whole number greater than 1 can be written as a product of prime numbers.

Example:
  • The number 12 can be written as 2 × 2 × 3.
  • So, the prime factorization of 12 is 22 × 3.
Note

Prime numbers have only two factors: 1 and the number itself. Factorization means breaking a number down into smaller factors until only prime numbers remain.

Understanding prime and composite numbers

A prime number has exactly two factors, 1 and itself. A composite number has more than two factors. Knowing the difference helps when finding prime factors.

Examples:
  • Prime numbers: 2, 3, 5, 7, 11, 13
  • Composite numbers: 4, 6, 8, 9, 10, 12
Note

The number 1 is neither prime nor composite because it has only one factor—1 itself.

How to find prime factors

To find the prime factors of a number, divide the number by the smallest prime number possible and continue dividing until all factors are prime.

Steps:
  • Start with the number 36.
  • Divide by 2 (the smallest prime): 36 ÷ 2 = 18.
  • Divide 18 by 2 again: 18 ÷ 2 = 9.
  • 9 is not divisible by 2, so divide by 3: 9 ÷ 3 = 3.
  • 3 is prime, so stop here.
  • The prime factors are 2 × 2 × 3 × 3, or 22 × 32.
Note

Always begin dividing by the smallest prime number. Continue until all remaining factors are prime numbers.

Using a factor tree

A factor tree is a diagram that shows how a number can be broken down into its prime factors step by step. Each branch splits until only prime numbers remain.

Example:
  • Start with 48.
  • Split into 6 × 8.
  • 6 = 2 × 3 and 8 = 2 × 4.
  • 4 = 2 × 2, so the prime factors are 2 × 2 × 2 × 2 × 3, or 24 × 3.
Note

A factor tree helps you organize your work and check that you found all the prime factors correctly.

Why prime factorization matters

Prime factorization is important because it helps in simplifying fractions, finding greatest common factors (GCF), and least common multiples (LCM).

Example:
  • To find the GCF of 18 and 24:
  • 18 = 2 × 32 and 24 = 23 × 3.
  • Common prime factors are 2 and 3.
  • Multiply the lowest powers: 21 × 31 = 6.
  • So, the GCF is 6.
Note

Understanding prime factorization gives you the tools to solve more complex math problems like finding GCFs and LCMs with confidence.