L3 · Prime factorization

Learning goal

“I can write the prime factorization of a composite number using a factor tree or the ladder method, and I know every correct tree ends at the same set of primes.”

Prime factorization — a multiplication made only of prime numbers that builds the number, like 12 = 2 × 2 × 3.

Factor tree — a diagram that keeps splitting composite numbers into factors until every branch ends in a prime.

Ladder method — repeated division by primes, written in a stack, until the quotient is 1.

Branch — one splitting step in a factor tree (36 splits into the branches 6 and 6).

Worked example

Prime factorization of 36. Tree: 36 → 6 × 6. Each 6 → 2 × 3. Every leaf is now prime: 2, 3, 2, 3. Collect and tidy: 36 = 2 × 2 × 3 × 3. Multiply back to check: 4 × 9 = 36. ✓ Say why it works: a branch only stops when it reaches a prime, because a composite can always be split once more.

Check yourself — explain it

A tree branch stops splitting when it reaches…

Paired problem — example still visible

Build a factor tree for 30 and write its prime factorization. → 30 = 2 × 3 × 5 (30 → 5 × 6 → 5 × (2 × 3))

Faded completion — last step is yours

“28 → 4 × 7; 4 splits into 2 × ___, so 28 = 2 × 2 × ___.” → 2; 7, so 28 = 2 × 2 × 7

Ready to practise?

10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.

Start practice →Take the unit quiz