GCF — Greatest Common Factor

The greatest common factor (GCF) of two numbers is the biggest number that divides both of them with no remainder.

Other names you'll see: greatest common divisor, GCD, highest common factor. Same thing.

Why it matters

GCF lets you simplify a fraction to lowest terms in one step instead of dividing top and bottom over and over.

12 / 18  →  GCF(12, 18) = 6  →  (12 ÷ 6) / (18 ÷ 6) = 2 / 3

If you don't use the GCF, you'd divide by 2, then by 3 — same answer, two steps. GCF makes it one step.

Two methods — pick the one that doesn't break for you

The listing method (slow, but visual)

1. List all factors of the first number.
2. List all factors of the second number.
3. The biggest number that appears in both lists is the GCF.

The trap: every number you write on a "factors of N" list must divide N exactly — test it. The most common mistake is mixing factors (numbers that divide into N) with multiples (numbers that N divides into). For example, the factors of 35 are 1, 5, 7, 35 — only those four. 10, 15, 20, 25, 30 are multiples of 5, not factors of 35. If you list those by mistake, you'll miss 7 and conclude the wrong answer.

If you're not sure, use Euclid's algorithm below — it doesn't care about lists.

Example 1: GCF(12, 18)

• Factors of 12: 1, 2, 3, 4, 6, 12
• Factors of 18: 1, 2, 3, 6, 9, 18
• Common: 1, 2, 3, 6
• GCF = 6

Example 2: GCF(15, 20)

• Factors of 15: 1, 3, 5, 15
• Factors of 20: 1, 2, 4, 5, 10, 20
• Common: 1, 5
• GCF = 5

Example 3: GCF(7, 9)

• Factors of 7: 1, 7 (7 is prime)
• Factors of 9: 1, 3, 9
• Common: 1
• GCF = 1 — these are coprime.

Faster way for two-digit numbers — Euclid's algorithm

This is the trick. Divide the bigger by the smaller, take the remainder, then divide the smaller by the remainder. Keep going. When the remainder is 0, the last non-zero remainder is the GCF.

Example: GCF(48, 18)

48 ÷ 18 = 2 remainder 12
18 ÷ 12 = 1 remainder 6
12 ÷ 6  = 2 remainder 0   ← stop
GCF = 6

Example: GCF(91, 26)

91 ÷ 26 = 3 remainder 13
26 ÷ 13 = 2 remainder 0   ← stop
GCF = 13

You can do this in your head once you see the pattern.

Things to remember

• GCF(a, 1) = 1 always.
• GCF(a, a) = a always.
• GCF(a, b) ≤ min(a, b) always — it can never be bigger than the smaller number.
• If one number divides the other, the smaller one is the GCF.
  • GCF(8, 24) = 8 (because 8 divides 24).

Try these (answers below)

1. GCF(9, 16)
2. GCF(20, 50)
3. GCF(36, 24)
4. GCF(7, 14)

1. 1 (9 and 16 share only the factor 1)
2. 10
3. 12
4. 7

Where this shows up

Look at the live tool at /fractions and switch to the GCF mode. You'll see factor lists exactly like the ones above.