Greatest common factor

Finding it by listing

Write out the factors of each number and take the biggest one they share.

Number	Factors
$12$	$1, 2, 3, 4, \mathbf{6}, 12$
$18$	$1, 2, 3, \mathbf{6}, 9, 18$

The shared factors are $1, 2, 3$ and $6$, so the GCF is $6$.

Finding it with primes

Break both numbers into primes and take everything they have in common.

$$12 = 2 \times 2 \times 3 \qquad 18 = 2 \times 3 \times 3$$

Both contain one $2$ and one $3$, so the GCF is $2 \times 3 = 6$. This method stays manageable when the numbers get large.

What it is for

Dividing a fraction’s top and bottom by their GCF puts it in simplest form in one step.

$$\frac{12}{18} = \frac{12 \div 6}{18 \div 6} = \frac{2}{3}$$

Dividing by a smaller shared factor still works — it just takes more steps. See simplifying fractions.

Not the same as the LCD

They pull in opposite directions, and mixing them up is a common slip.

	Meaning	Used for
GCF	largest number that divides both	making fractions smaller
LCM or LCD	smallest number both divide into	giving fractions a shared bottom

For $4$ and $6$ the GCF is $2$ and the least common denominator is $12$.

When the GCF is 1

Two numbers whose only shared factor is $1$ are called coprime. A fraction built from them, like $\tfrac{5}{9}$, is already in simplest form.