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Factors and Multiples
Understanding factors and multiples through their uses and differences
What are factors?
What are multiples?
How are they related and different?
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What if....
You have these two planks of wood, and you want to cut them into smaller planks with the same length.
What are the possible measurements of the shorter planks?
36 inches
24 inches
What are factors?
A factor is a number that can be used to divide another number without any remainder.
6 is a factor of 36
Can you think of other factors of 36?
Finding all the factors
The factors of a number are those pairs or sets of numbers that, when multiplied, result in the number.
6 × 6 =
9 × 4 =
12 × 3 =
18 × 2 =
9 × 2 × 2 =
36 × 1 =
36
The factors of 36 are
1 , 2 , 3, 4, 6, 9, 12, 18 and 36.
Exercise:
What are the factors of 24?
24
= ? × ?
= ? × ?
= ? × ?
= ? × ? × ?
= ? × ? × ?
We can solve the problem earlier using the factors of 36.
36 inches
24 inches
Exercise (Answer Key):
What are the factors of 24?
1 × 24 =
2 × 12 =
3 × 8 =
4 × 6 =
24
= 2 × 2 × 6
= 4 × 3 × 2
= 2 × 2 × 2 × 3
Did you know?
We can express repeating prime factors in their exponential form!
For example, we can rewrite
2 × 2 × 2 × 3 to 2³ × 3.
The factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24.
Notice that the numbers are only listed once.
Solving Problems Using Factors
Length of Planks | Factors |
36 | 1, 2, 3, 4, 6, 9, 12, 18 and 36. |
24 | 1, 2, 3, 4, 6, 8, 12 and 24. |
Common Factors: 1, 2, 3, 4, 6 and 12
We can cut the planks using any of the common factors as lengths to have equal measurements for the pieces.
36 inches
24 inches
- How many planks will you have if you cut the two into 4-inch planks?
36 inches
4
4
4
4
.........
- We can solve this problem by finding out the multiples of 4.
What are multiples?
A multiple is the product of a number and any integer.
4 × 1 = 4
4 × 2 = 8
4 × 3 = 12
4 × 4 = 16
4, 8, 12 and 16 are some of the multiples of 4
Recall:
Integers are whole numbers that can be positive, negative or zero. But note that the examples will be focusing on positive integers.
Finding multiples
Let’s find out the multiples of a number. We may start multiplying it by 1, then by 2 and so on.
36 inches
4 × 9 = 36
4 × 8 = 32
4 × 7 = 28
4 × 6 = 24
4 × 5 = 20
4 × 4 = 16
4 × 3 = 12
4 × 2 = 8
4 × 1 = 4
For this item, we need to multiply 4 until we reach 36 to know the number of 4-inch planks that will be made.
Finding multiples
Using the concept of multiples, how many 4-inch planks will we get from the 24-inch plank?
4
1
8
2
12
3
16
4
20
5
24
6
24 inches
We can get six 4-inch planks.
Can you think of the first 15 multiples of 4?
Answer Key:
How many planks will you have if you cut the two into 4-inch planks?
Answer: 15
This can be checked using the 15th multiple of 4 and comparing it with the total length of the 2 planks.
36 inches
4
24 inches
First 15 multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60
Total length: 36 + 24 = 60
Factors and Multiples
How are they connected and different from each other?
4 × 9 = 36
A factor can divide another number without a remainder.
A multiple can be divided by another number without a remainder.
36 ÷ 6 = 6
36 ÷ 9 = 4
Resources
Want to learn more?
Review '2.06 Factors and Multiples' from Mathspace.
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