Are GCF, HCF and GCD the same thing?

Yes. Greatest common factor, highest common factor and greatest common divisor are three names for the same number.

What is the GCF of two primes?

If the primes are different, their only shared factor is 1, so the GCF is 1 and the LCM is their product.

Greatest Common Factor and Least Common Multiple (GCF and LCM)

How to find the GCF and LCM by listing and by prime factors, a Venn diagram way to see both at once, and where each one turns up: simplifying fractions and finding common denominators.

3 min read Supports 4 lessons

Two numbers usually share some factors and some multiples. The GCF is the biggest factor they share. The LCM is the smallest multiple they share. One looks down into the numbers; the other looks up from them.

By listing

For small numbers, listing works well. Take $12$ and $18$ .

\begin{aligned}\text{factors of } 12 &: 1, 2, 3, 4, \mathbf{6}, 12 \\ \text{factors of } 18 &: 1, 2, 3, \mathbf{6}, 9, 18\end{aligned}

The largest shared factor is $6$ , so $\gcd(12, 18) = 6$ .

\begin{aligned}\text{multiples of } 12 &: 12, 24, \mathbf{36}, 48, \dots \\ \text{multiples of } 18 &: 18, \mathbf{36}, 54, \dots\end{aligned}

The first shared multiple is $36$ , so $\operatorname{lcm}(12, 18) = 36$ .

By prime factors

For bigger numbers, write each as a product of primes. Then the GCF takes the primes they share, at the lowest power, and the LCM takes every prime, at the highest power.

$12 = 2^2 \cdot 3 \qquad 18 = 2 \cdot 3^2$
Write both numbers as prime factorisations.

1/5

Tip

Picture a Venn diagram of prime factors. $12$ has $2, 2, 3$ and $18$ has $2, 3, 3$ . The overlap holds $2, 3$ : multiply those for the GCF. Everything in the diagram, $2, 2, 3, 3$ , multiplies to the LCM.

A neat check

Product rule

\gcd(a,b) \times \operatorname{lcm}(a,b) = a \times b

For $12$ and $18$ : $6 \times 36 = 216 = 12 \times 18$ .

This works because the GCF and LCM between them use every prime factor of $a$ and $b$ exactly once.

Where they are used

The GCF simplifies. Dividing the top and bottom of $\frac{12}{18}$ by their GCF, $6$ , gives $\frac{2}{3}$ in one step. In algebra, factoring out the GCF is the first move in factorising an expression.

The LCM combines. To add $\frac{1}{12} + \frac{1}{18}$ you need a common denominator, and the LCM, $36$ , is the smallest one.

Worked example

Buses leave a station every $12$ minutes and trains every $18$ minutes. Both leave at 9:00. When do they next leave together?

1
They coincide at shared multiples of $12$ and $18$ .

Watch out

Do not mix them up. If the answer is smaller than both numbers, it is a factor (GCF). If it is bigger than both, it is a multiple (LCM).

Common questions

Are GCF, HCF and GCD the same thing?: Yes. Greatest common factor, highest common factor and greatest common divisor are three names for the same number.
What is the GCF of two primes?: If the primes are different, their only shared factor is $1$ , so the GCF is $1$ and the LCM is their product.

Practise gcf and lcm

The GCF & LCM lesson has worked examples and unlimited practice that adapts to you.

Open the lesson

Number & Arithmetic

Greatest Common Factor and Least Common Multiple (GCF and LCM)

How to find the GCF and LCM by listing and by prime factors, a Venn diagram way to see both at once, and where each one turns up: simplifying fractions and finding common denominators.

3 min read Supports 4 lessons

By listing

For small numbers, listing works well. Take $12$ and $18$ .

\begin{aligned}\text{factors of } 12 &: 1, 2, 3, 4, \mathbf{6}, 12 \\ \text{factors of } 18 &: 1, 2, 3, \mathbf{6}, 9, 18\end{aligned}

The largest shared factor is $6$ , so $\gcd(12, 18) = 6$ .

\begin{aligned}\text{multiples of } 12 &: 12, 24, \mathbf{36}, 48, \dots \\ \text{multiples of } 18 &: 18, \mathbf{36}, 54, \dots\end{aligned}

The first shared multiple is $36$ , so $\operatorname{lcm}(12, 18) = 36$ .

By prime factors

For bigger numbers, write each as a product of primes. Then the GCF takes the primes they share, at the lowest power, and the LCM takes every prime, at the highest power.

$12 = 2^2 \cdot 3 \qquad 18 = 2 \cdot 3^2$
Write both numbers as prime factorisations.

1/5

Tip

A neat check

Product rule

\gcd(a,b) \times \operatorname{lcm}(a,b) = a \times b

For $12$ and $18$ : $6 \times 36 = 216 = 12 \times 18$ .

This works because the GCF and LCM between them use every prime factor of $a$ and $b$ exactly once.

Where they are used

The LCM combines. To add $\frac{1}{12} + \frac{1}{18}$ you need a common denominator, and the LCM, $36$ , is the smallest one.

Worked example

Buses leave a station every $12$ minutes and trains every $18$ minutes. Both leave at 9:00. When do they next leave together?

1
They coincide at shared multiples of $12$ and $18$ .

Watch out

Do not mix them up. If the answer is smaller than both numbers, it is a factor (GCF). If it is bigger than both, it is a multiple (LCM).

Common questions

Are GCF, HCF and GCD the same thing?: Yes. Greatest common factor, highest common factor and greatest common divisor are three names for the same number.
What is the GCF of two primes?: If the primes are different, their only shared factor is $1$ , so the GCF is $1$ and the LCM is their product.

Practise gcf and lcm

The GCF & LCM lesson has worked examples and unlimited practice that adapts to you.

Open the lesson