Factors and Multiples

Discover what factors and multiples are, how to list them, and how to find common factors and multiples of two numbers.

What Are Factors?

A factor of a number is a whole number that divides into it exactly, with no remainder.

For example, the factors of $12$ are: $1, 2, 3, 4, 6, 12$

A quick way to find all factors is to look for factor pairs — two numbers that multiply to give the original number:

$1 \times 12 = 12, \quad 2 \times 6 = 12, \quad 3 \times 4 = 12$

Note

Every number has at least two factors: $1$ and itself. A number with exactly two factors is called a prime number!

What Are Multiples?

A multiple of a number is what you get when you multiply it by any whole number.

Multiples of $5$ : $5, 10, 15, 20, 25, 30, \ldots$

Multiples of $3$ : $3, 6, 9, 12, 15, 18, 21, 24, \ldots$

You can always find more multiples — the list goes on forever.

Highest Common Factor (HCF)

The HCF of two numbers is the largest factor they share.

Example: Find the HCF of $18$ and $24$ .

Factors of $18$ : $1, 2, 3, 6, 9, 18$
Factors of $24$ : $1, 2, 3, 4, 6, 8, 12, 24$
Common factors: $1, 2, 3, 6$
HCF = 6

Tip

The HCF is useful for simplifying fractions. To simplify $\frac{18}{24}$ , divide both numbers by HCF $= 6$ to get $\frac{3}{4}$ .

Lowest Common Multiple (LCM)

The LCM of two numbers is the smallest multiple they share.

Example: Find the LCM of $4$ and $6$ .

Multiples of $4$ : $4, 8, \mathbf{12}, 16, 20, 24, \ldots$
Multiples of $6$ : $6, \mathbf{12}, 18, 24, \ldots$
LCM = 12

The LCM is useful for adding fractions with different denominators.

Warning

Don't mix up HCF and LCM. HCF is the biggest shared factor (so it is smaller than or equal to the numbers). LCM is the smallest shared multiple (so it is larger than or equal to the numbers).

Real world: event scheduling with LCM

A traffic light changes red every 3 minutes and amber every 4 minutes. Both change at the same time when $\text{LCM}(3, 4) = 12$ minutes have elapsed. Transport planners use LCM to synchronise traffic light sequences and minimise congestion.

Real world: simplifying fractions in baking

A baker uses $\frac{18}{24}$ cup of sugar. $\text{HCF}(18, 24) = 6$ , so $\frac{18}{24} = \frac{3}{4}$ cup. Using the HCF to simplify fractions is an everyday skill in recipes and measurement.

Real world: packaging equal bundles

A printer has 48 red leaflets and 60 blue leaflets to bundle into equal sets with no leftovers. $\text{HCF}(48, 60) = 12$ , so 12 identical bundles can be made, each with 4 red and 5 blue leaflets. The HCF gives the maximum number of equal bundles.

Tip Remember: HCF (highest common factor) is always ≤ both numbers; LCM (lowest common multiple) is always ≥ both numbers. The product of HCF and LCM equals the product of the two numbers.

Practice Problems

List all factors of 20.

Show answer

Factor pairs: $1 \times 20$, $2 \times 10$, $4 \times 5$. Factors: $1, 2, 4, 5, 10, 20$.

Find the LCM of 3 and 8.

Show answer

Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24... Multiples of 8: 8, 16, 24... LCM = 24.