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Learn Index · Number Sense and Numeration Grade 6–8

Prime factorization

Writing a number as a product of primes.

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Breaking numbers into primes

Prime factorization means writing a composite number as a product of prime numbers only.

  • 18 = 2 × 3 × 3 = 2 × 3²
  • 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5

The exponent (small raised number) shows how many times a prime is used.

Primes are the building blocks of all whole numbers, the way atoms are building blocks of matter.

One answer only

The Fundamental Theorem of Arithmetic says every whole number greater than 1 has exactly one prime factorization (apart from the order). However you start, you’ll end up with the same primes.

Method 1: Factor tree

  1. Write the number at the top.
  2. Split it into any two factors.
  3. Keep splitting each factor that isn’t prime.
  4. Circle the primes at the ends of the branches.
  5. Multiply all the circled primes.

Example: 72

  • 72 = 8 × 9
  • 8 = 2 × 4, and 4 = 2 × 2
  • 9 = 3 × 3
  • Primes: 2, 2, 2, 3, 3

72 = 2³ × 3²

Starting with 72 = 6 × 12 gives the same result.

Method 2: Repeated division

Divide by the smallest prime that works, over and over:

Divide by Result
2 120 ÷ 2 = 60
2 60 ÷ 2 = 30
2 30 ÷ 2 = 15
3 15 ÷ 3 = 5
5 5 ÷ 5 = 1

120 = 2³ × 3 × 5

Why it’s useful

  • Greatest common factor: multiply the primes both numbers share.
  • Least common multiple: multiply the highest power of every prime that appears.
  • Simplifying fractions and square roots.
  • Checking for perfect squares: all exponents are even. 36 = 2² × 3² is a perfect square.

Worked examples

Example 1: Prime factorization of 18.

18 = 2 × 9 = 2 × 3 × 3 = 2 × 3².

Example 2: Prime factorization of 100.

100 = 10 × 10 = 2 × 5 × 2 × 5 = 2² × 5².

Example 3: Prime factorization of 84.

84 = 2 × 42 = 2 × 2 × 21 = 2 × 2 × 3 × 7 = 2² × 3 × 7.

Common mistakes

  • Stopping too early, at 3 × 6 where 6 isn’t prime.
  • Including 1 as a factor. 1 is not prime.
  • Forgetting a factor when writing the final answer. Multiply back to check.

Try it yourself

  1. 24
  2. 45
  3. 150
  4. 98
  5. Is 2² × 3³ a perfect square?

Answers: 1) 2³ × 3 2) 3² × 5 3) 2 × 3 × 5² 4) 2 × 7² 5) No: the exponent 3 is odd.

Quick recap

  • Prime factorization writes a number as a product of primes.
  • Use a factor tree or repeated division.
  • Every number has exactly one prime factorization.