Factors and Multiples

What you will learn

• List all the factors of a number by finding pairs.
• List multiples of a number.
• Use divisibility rules to test for factors quickly.

Before you start: you should know your times tables and division. See Division: Sharing and Grouping.

The idea

If $3 \times 4 = 12$, then $3$ and $4$ are factors of $12$, and $12$ is a multiple of both $3$ and $4$. The two words describe the same multiplication fact from two directions.

Factors

A factor of a number divides into it exactly, with no remainder. To find all the factors of a number, look for pairs that multiply to it. Start at $1$ and work up.

Stop when the pairs start to repeat. For $12$, after $3 \times 4$ the next candidate is $4$, which you have already found.

Multiples

A multiple of a number is what you get when you multiply it by $1, 2, 3, 4, \dots$. In other words, it is the numbers in its times table, and counting in steps lists them.

Multiples go on forever, but a number has only a few factors, and the biggest is the number itself.

Divisibility tests

You can tell if a number is a multiple of another without dividing.

• By $2$: it ends in $0, 2, 4, 6$ or $8$.
• By $5$: it ends in $0$ or $5$.
• By $10$: it ends in $0$.
• By $3$: the digits add to a multiple of $3$. For $126$: $1 + 2 + 6 = 9$, so yes.
• By $9$: the digits add to a multiple of $9$.

Factors and multiples together

If $a$ is a factor of $b$, then $b$ is a multiple of $a$. Every number is a factor and a multiple of itself.

Worked examples

Example 1: All the factors

Find all the factors of $30$.

1. $1 \times 30$.
2. $2 \times 15$.
3. $3 \times 10$.
4. $5 \times 6$. The next number to try is $6$, already found.

Answer: $1, 2, 3, 5, 6, 10, 15, 30$.

Example 2: Is it a multiple?

Is $138$ a multiple of $3$?

1. Add the digits: $1 + 3 + 8 = 12$.
2. $12$ is a multiple of $3$.

Answer: Yes. $138 = 3 \times 46$.

Example 3: Multiples in context

Buses leave a stop every $15$ minutes starting at $9\!:\!00$. When is the fourth bus?

1. Count in steps of $15$ minutes from the start: the buses leave at $9\!:\!00, 9\!:\!15, 9\!:\!30, 9\!:\!45$.
2. The fourth bus leaves at $9\!:\!45$.

Answer: $9\!:\!45$.

Do not mix up the two words. Factors are never bigger than the number. Multiples are never smaller than it.

Why it matters

Factors let you share things equally and simplify fractions. Multiples are what you use for timetables, patterns and finding when two cycles meet, which you will see in Highest Common Factor and Lowest Common Multiple.