Prime Factorization Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Prime Factorization.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Writing a whole number as a product of prime numbers; every composite number has exactly one such representation (up to order).

Break a number into building blocks that cannot be split further (primes).

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Every composite number breaks down into a unique set of prime factors—this uniqueness is the Fundamental Theorem of Arithmetic.

Common stuck point: Students stop factor trees too early at composite leaves—every branch must end at a prime; check each factor.

Sense of Study hint: Continue splitting until every factor is prime, then sort factors.

Worked Examples

Example 1

easy
Find the prime factorization of 360 using a factor tree.

Solution

  1. 1
    Start: 360 = 2 \times 180.
  2. 2
    180 = 2 \times 90; 90 = 2 \times 45; 45 = 3 \times 15; 15 = 3 \times 5.
  3. 3
    Collect all prime factors: 2 \times 2 \times 2 \times 3 \times 3 \times 5 = 2^3 \times 3^2 \times 5.
  4. 4
    Verify: 8 \times 9 \times 5 = 72 \times 5 = 360. ✓

Answer

360 = 2^3 \times 3^2 \times 5
Prime factorization expresses any composite number as a product of prime numbers. The result is unique by the Fundamental Theorem of Arithmetic — no matter what order you factor, the prime factors and their exponents are always the same.

Example 2

hard
Use prime factorization to find \gcd(180, 252) and \text{lcm}(180, 252).

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Write the prime factorization of 84 and use it to find all of its factors.

Example 2

medium
A rectangular garden can be arranged as a whole-number rectangle in exactly 6 different ways (including 1 \times n and n \times 1 as different). What is the smallest possible area for this garden?
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Background Knowledge

These ideas may be useful before you work through the harder examples.

prime numberscomposite numbersdivisibility intuition