Factors and Multiples | Grade 5 Math | LevelKite

Sometimes a group of things splits into equal rows with nothing left over. Sometimes a few are left behind. In this lesson you will learn two words that describe this: factor and multiple.

One note first. This lesson uses counting numbers. These are the whole numbers you count with: $1$ , $2$ , $3$ , $4$ and so on. Zero is not one of them.

Start with $12$ counters. Put them into $3$ equal rows. Each row gets $4$ counters, and none are left over. As a number sentence, that is $3 \times 4 = 12$ .

Now try $5$ equal rows. Each row gets $2$ counters, but $2$ counters are left over. In numbers, $5 \times 2 = 10$ , and $2$ more make $12$ . Twelve counters cannot be put into $5$ equal rows without leftovers.

A factor of a number is a counting number that divides it with no remainder. No remainder means nothing is left over, so the remainder is $0$ .

Look at the counters again, this time as division.

$12 \div 3 = 4$ with remainder $0$ . So $3$ is a factor of $12$ .
$12 \div 5 = 2$ with remainder $2$ . So $5$ is not a factor of $12$ .

That gives you a test that works for any counting numbers. To check whether one number is a factor of another, divide the other number by it. A remainder of $0$ means it is a factor. Any other remainder means it is not.

Worked example

Testing for a factor

Is $7$ a factor of $91$ ? Is $7$ a factor of $95$ ?

A factor must divide the number with no remainder. So divide 91 by 7 and look at the remainder.
$91 \div 7 = 13 \text{ remainder } 0$
Check the division by multiplying back. Thirteen groups of 7 make exactly 91, so nothing is left over. The remainder is 0, which means 7 is a factor of 91.
$13 \times 7 = 91$
Use the same test on 95. Divide 95 by 7 and look at the remainder.
$95 \div 7 = 13 \text{ remainder } 4$
Check by multiplying back. Thirteen groups of 7 make 91, and 4 more are needed to reach 95. Those 4 are left over, so the remainder is not 0. That means 7 is not a factor of 95.
$13 \times 7 + 4 = 95$

Answer

Yes, $7$ is a factor of $91$ , because $91 \div 7$ leaves remainder $0$ . No, $7$ is not a factor of $95$ , because $95 \div 7$ leaves remainder $4$ .

Now turn the idea around. Instead of splitting a number into equal groups, build numbers out of equal groups.

A multiple of a number is the answer you get when you multiply that number by a counting number. You make multiples every time you skip count. Skip counting by fours gives $4$ , $8$ , $12$ , $16$ , $20$ . Each of those is a multiple of $4$ . The list starts with $4$ itself, because $1$ group of $4$ is just $4$ .

The table shows these multiples three ways: in words, as a number sentence, and as the multiple itself.

The first five multiples of 4

In words	Number sentence	Multiple of 4
1 group of 4	$1 \times 4 = 4$	$4$
2 groups of 4	$2 \times 4 = 8$	$8$
3 groups of 4	$3 \times 4 = 12$	$12$
4 groups of 4	$4 \times 4 = 16$	$16$
5 groups of 4	$5 \times 4 = 20$	$20$

Look at the third row. Those $3$ groups of $4$ are the same $3$ rows of $4$ counters you started with. So $12$ is a multiple of $4$ .

Skip counting works for small numbers, but it is slow for big ones. Division is quicker. A number is a multiple of $4$ when it splits into groups of $4$ with nothing left over.

$36 \div 4 = 9$ with remainder $0$ . So $36 = 9 \times 4$ , and $36$ is a multiple of $4$ .
$38 \div 4 = 9$ with remainder $2$ . So $38$ is not a multiple of $4$ .

The same test works with any counting number in place of $4$ . Divide by it. A remainder of $0$ means the number you divided is one of its multiples. Any other remainder means it is not.

Worked example

Listing and testing multiples

List the first five multiples of $6$ . Then decide whether $84$ is a multiple of $6$ , and whether $100$ is a multiple of $6$ .

Multiply 6 by the counting numbers 1, 2, 3, 4 and 5, in order. Each answer is a multiple of 6, because it is 6 multiplied by a counting number.
$1 \times 6 = 6 \quad 2 \times 6 = 12 \quad 3 \times 6 = 18 \quad 4 \times 6 = 24 \quad 5 \times 6 = 30$
Skip counting by sixes all the way to 84 would be slow, so divide instead. If 84 splits into groups of 6 with nothing left over, it is a multiple of 6.
$84 \div 6 = 14 \text{ remainder } 0$
The remainder is 0, so 84 is exactly 14 groups of 6. That makes 84 a multiple of 6.
$14 \times 6 = 84$
Test 100 the same way. The remainder is 4, so 100 does not split into groups of 6 with nothing left over. It falls between two multiples of 6: 16 groups of 6 make 96, and 17 groups of 6 make 102.
$100 \div 6 = 16 \text{ remainder } 4$

Answer

The first five multiples of $6$ are $6$ , $12$ , $18$ , $24$ and $30$ . The number $84$ is a multiple of $6$ , because $84 = 14 \times 6$ . The number $100$ is not a multiple of $6$ , because $100 \div 6$ leaves remainder $4$ .

Key concept

One multiplication fact, two words

Take any multiplication fact made of counting numbers, such as $3 \times 4 = 12$ .

The numbers you multiply, $3$ and $4$ , are factors of $12$ .
The answer, $12$ , is a multiple of $3$ and a multiple of $4$ .

So 3 is a factor of 12 and 12 is a multiple of 3 say the same thing. One division checks both: $12 \div 3 = 4$ with remainder $0$ .

\text{factor} \times \text{factor} = \text{multiple}

Factors and multiples of the same number behave very differently. Take the number $6$ .

You already know how its multiples start: $6$ , $12$ , $18$ , $24$ , $30$ . To find its factors, test every counting number from $1$ up to $6$ . The table shows each test.

Which counting numbers from 1 to 6 are factors of 6?

Number tested	Divide 6 by it	Factor of 6?
$1$	$6 \div 1 = 6$ , remainder $0$	Yes
$2$	$6 \div 2 = 3$ , remainder $0$	Yes
$3$	$6 \div 3 = 2$ , remainder $0$	Yes
$4$	$6 \div 4 = 1$ , remainder $2$	No
$5$	$6 \div 5 = 1$ , remainder $1$	No
$6$	$6 \div 6 = 1$ , remainder $0$	Yes

Only $1$ , $2$ , $3$ and $6$ pass the test. What about a number greater than $6$ , such as $7$ ? Seven does not fit into six even once, so $6 \div 7 = 0$ with remainder $6$ . All $6$ are left over. The same thing happens with every number greater than $6$ . So the list of factors is complete: $1$ , $2$ , $3$ , $6$ .

The multiples of $6$ are different. After $6$ , $12$ , $18$ , $24$ and $30$ come $36$ , $42$ , $48$ and more. You can always multiply $6$ by the next counting number, and the counting numbers never run out. So the list of multiples never ends.

The same is true for every counting number:

Its factors must divide it with no remainder, so none is greater than the number. That means there are only so many of them.
Its multiples come from multiplying it by $1$ , $2$ , $3$ and so on. So none is less than the number, and they go on without end.
The number itself is on both lists. It is its own greatest factor and its own first multiple.

Worked example

Factor, multiple, both, or neither?

Look at these five numbers: $2$ , $6$ , $8$ , $72$ and $100$ . Decide whether each one is a factor of $8$ , a multiple of $8$ , both, or neither.

The numbers 2 and 6 are less than 8. They cannot be multiples of 8, because the first multiple of 8 is 8 itself. So test them only as factors: divide 8 by each one. The remainder for 2 is 0, so 2 is a factor of 8. The remainder for 6 is 2, so 6 is not a factor. It is neither.
$8 \div 2 = 4 \text{ remainder } 0 \qquad 8 \div 6 = 1 \text{ remainder } 2$
The number 8 divides itself exactly once, and 1 group of 8 is 8. So 8 is both a factor of 8 and a multiple of 8.
$8 \div 8 = 1 \text{ remainder } 0 \qquad 1 \times 8 = 8$
The numbers 72 and 100 are greater than 8. They cannot be factors of 8, because a factor of a counting number is never greater than that number. So test them only as multiples: divide each one by 8. The remainder for 72 is 0, so 72 is a multiple of 8. The remainder for 100 is 4, so 100 is not a multiple. It is neither.
$72 \div 8 = 9 \text{ remainder } 0 \qquad 100 \div 8 = 12 \text{ remainder } 4$

Answer

Of the five numbers, $2$ is a factor of $8$ , $72$ is a multiple of $8$ , and $8$ is both. The numbers $6$ and $100$ are neither a factor nor a multiple of $8$ .

Here is the idea to keep. A factor of a counting number divides it with no remainder. A multiple of a counting number is that number times a counting number. One multiplication fact shows both: in $3 \times 4 = 12$ , the numbers $3$ and $4$ are factors of $12$ , and $12$ is a multiple of each. A counting number has only so many factors, and none is greater than the number. Its multiples start at the number itself and go on without end.

Next, you will learn quick rules that tell you whether $2$ , $3$ , $5$ , $6$ , $9$ or $10$ is a factor of a number. You will also learn a faster way to find all the factors of a number, and you will sort numbers by how many factors they have.