Factors and multiples

A factor of $n$ is any whole number that divides $n$ exactly (with no remainder). A multiple of $n$ is any number obtained by multiplying $n$ by a positive integer.

Idea

For example, the factors of $12$ are: $1, 2, 3, 4, 6, 12$. The multiples of $12$ are: $12, 24, 36, 48, 60, \dots$.

A few facts:

• $1$ is a factor of every number.
• Every number is a factor of itself.
• The number of factors of $n$ is finite.
• The number of multiples of $n$ is infinite.
• $0$ is a multiple of every number (since $0 = 0 \cdot n$), but not a factor of any number.

Divisibility tests (you should memorise the easy ones):

These tests turn long divisions into quick mental checks.

Worked examples

Example 1. List all factors of $24$.

$24 = 1 \times 24 = 2 \times 12 = 3 \times 8 = 4 \times 6$. So factors: $1, 2, 3, 4, 6, 8, 12, 24$.

Example 2. Is $1{,}248$ divisible by $4$?

Last two digits: $48$. $48 \div 4 = 12$, no remainder. So yes.

Example 3. Is $531$ divisible by $9$?

Sum of digits: $5 + 3 + 1 = 9$. Divisible by $9$. So yes.

Example 4. First five multiples of $7$.

$7, 14, 21, 28, 35$.

Try it yourself

1. List all factors of $36$.
2. Is $4567$ divisible by $3$?
3. Is $2310$ divisible by $6$?
4. First $5$ multiples of $9$.
5. How many factors does $48$ have?
6. Test if $87654$ is divisible by $9$ (sum digits).
7. Find a number less than $50$ with exactly $5$ factors.
8. Is $0$ a factor of $20$? Why or why not?

Activity

Pick any number between $1$ and $100$. Find all its factors and the first ten multiples. Share with a partner , they verify.