L4 · Common factors from prime factorization

Learning goal

“I can find all the common factors of two numbers — and the greatest one — by comparing their prime factorizations.”

Common factor — a factor that two (or more) numbers share. 3 is a common factor of 12 and 18.

Greatest common factor (GCF) — the largest of the common factors. The GCF of 12 and 18 is 6.

Shared prime — a prime factor that appears in both numbers' prime factorizations, like the 2 and a 3 shared by 12 = 2 × 2 × 3 and 18 = 2 × 3 × 3.

Worked example

Common factors of 12 and 18. Prime factorizations side by side: 12 = 2 × 2 × 3; 18 = 2 × 3 × 3. Circle the shared primes: one 2 and one 3 appear in both. Common factors are built from the shared primes: 2, 3, and 2 × 3 = 6 — plus 1, always. So the common factors are 1, 2, 3, 6, and the greatest is 6. Confirm with factor rainbows: 12 → 1, 2, 3, 4, 6, 12; 18 → 1, 2, 3, 6, 9, 18 — the overlap is exactly 1, 2, 3, 6. Say why it works: a common factor's primes must live inside both numbers, so they can only come from the shared primes.

Check yourself — explain it

Why is 4 a factor of 12 but NOT a common factor of 12 and 18?

Paired problem — example still visible

Find the GCF of 10 and 15 using prime factorizations. → 10 = 2 × 5, 15 = 3 × 5; shared prime is 5, so GCF = 5

Faded completion — last step is yours

“20 = 2 × 2 × 5 and 30 = 2 × 3 × 5. Shared primes: 2 and 5. GCF = 2 × 5 = ___.” → 10

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

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