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Lesson 1: Factors and multiples

# Factors and multiples

Learn about factors and multiples and how they relate to each other.

## Factors

Factors are whole numbers that can evenly divide another number.

### Picturing factors

Factors give us a way to break down a number into smaller pieces. We can arrange dots into equal sized groups to help us picture the factors of 12.
12 dots can be arranged in start color #01d1c1, 1, end color #01d1c1 row with start color #01d1c1, 12, end color #01d1c1 dots.
start color #01d1c1, 1, times, 12, end color #01d1c1, equals, 12
An array of 12 circles arranged in 1 row of 12 circles.
12 dots can also be arranged in start color #aa87ff, 2, end color #aa87ff rows with start color #aa87ff, 6, end color #aa87ff dots per row.
start color #aa87ff, 2, times, 6, end color #aa87ff, equals, 12
An array of 12 circles arranged in 2 rows of 6 circles each.
Or, we can arrange 12 dots in start color #f9685d, 3, end color #f9685d rows with start color #f9685d, 4, end color #f9685d dots in each row.
start color #f9685d, 3, times, 4, end color #f9685d, equals, 12
An array of 12 circles arranged in 3 rows of 4 circles each.
Once we figure out all of the ways that 12 dots can be arranged, we can look at the number of rows and number of dots in each row to find the factors of 12.
start color #01d1c1, 1, end color #01d1c1, start color #01d1c1, 12, end color #01d1c1, start color #aa87ff, 2, end color #aa87ff, start color #aa87ff, 6, end color #aa87ff, start color #f9685d, 3, end color #f9685d, and start color #f9685d, 4, end color #f9685d are all factors of 12.
We can make 12 with a row of start color #1fab54, 5, end color #1fab54 and a row of start color #e07d10, 7, end color #e07d10. So are start color #1fab54, 5, end color #1fab54 and start color #e07d10, 7, end color #e07d10 factors of 12?
An array of 12 circles arranges in 1 row of 5 circles and 1 row of 7 circles.
No. start color #1fab54, 5, end color #1fab54 and start color #e07d10, 7, end color #e07d10 are not factors because the dots are not divided into equal sized groups.
Which of the following arrangements are possible for 18 dots?

So what are the factors of 18?

### Finding factors without pictures

We can find the factors for 16 without drawing dots by thinking about numbers that will divide into 16 evenly.
start color #11accd, 1, end color #11accd is a factor of 16 because start color #11accd, 1, end color #11accd can be divided into 16 with no remainder.
16, divided by, start color #11accd, 1, end color #11accd, equals, start color #11accd, 16, end color #11accd
The quotient, which is start color #11accd, 16, end color #11accd, comma is also a factor of 16.
start color #1fab54, 2, end color #1fab54 is a factor of 16 because start color #1fab54, 2, end color #1fab54 can be divided into 16 with no remainder.
16, divided by, start color #1fab54, 2, end color #1fab54, equals, start color #1fab54, 8, end color #1fab54
The quotient, which is start color #1fab54, 8, end color #1fab54, is also a factor of 16.
start color #7854ab, 4, end color #7854ab is a factor of 16 because start color #7854ab, 4, end color #7854ab can be divided into 16 with no remainder.
16, divided by, start color #7854ab, 4, end color #7854ab, equals, start color #7854ab, 4, end color #7854ab
In this case the quotient is start color #7854ab, 4, end color #7854ab, which we have already discovered is a factor of 16.
The factors of 16 are start color #11accd, 1, comma, 16, end color #11accd, start color #1fab54, 2, comma, 8, end color #1fab54, comma and start color #7854ab, 4, end color #7854ab.
Numbers like 3 and 5 are not factors of 16 because they cannot be divided evenly into 16.
Use division to determine which of the following numbers are factors of 35.
Factor $1$1 $2$2 $3$3 $5$5 $7$7 $35$35

## Factor hints

Every number has 1 as a factor.
1 is a factor of 10.
1 is a factor of 364.
1 is a factor of 5, comma, 787.
Every number has itself as a factor.
41 is a factor of 41.
128 is a factor of 128.
4, comma, 379 is a factor of 4, comma, 379.

## Factor pairs

Two numbers that we multiply together to get a certain product are called factor pairs. To get the product of 8, we can multiply start color #543b78, 1, end color #543b78 times start color #543b78, 8, end color #543b78 and start color #208170, 2, end color #208170 times start color #208170, 4, end color #208170. So the factor pairs for 8 are start color #543b78, 1, end color #543b78 and start color #543b78, 8, end color #543b78 and start color #208170, 2, end color #208170 and start color #208170, 4, end color #208170.
Arranging dots in equal sized groups helps us to see that factors always come in pairs. One factor in the factor pair is the number of rows. The other factor in the factor pair is the number of dots in each row.
An array of 8 circles arranged in 1 row of 8 circles.
An array of 8 circles arranged in 2 rows of 8 circles each.
Let's find the factor pairs of 20. Remember, we are looking for two whole numbers that we can multiply together to get 20.
We'll start with start color #0c7f99, 1, end color #0c7f99 because we know that start color #0c7f99, 1, end color #0c7f99 is a factor for every number. We multiply start color #0c7f99, 1, end color #0c7f99, times, start color #0c7f99, 20, end color #0c7f99, to get 20, so start color #0c7f99, 20, end color #0c7f99 is also a factor. We can list these factors as the outside ends of a list, leaving room in the middle for additional factors.
start color #0c7f99, 1, end color #0c7f99start color #0c7f99, 20, end color #0c7f99
Now we check to see whether the next counting number, 2, is a factor.
Is there a whole number we can multiply by start color #208170, 2, end color #208170 to get 20? Yes. start color #208170, 2, end color #208170, times, start color #208170, 10, end color #208170, equals, 20. So start color #208170, 2, end color #208170 and start color #208170, 10, end color #208170 are another factor pair.
start color #0c7f99, 1, end color #0c7f99start color #208170, 2, end color #208170start color #208170, 10, end color #208170start color #0c7f99, 20, end color #0c7f99
The next counting number is 3. Is there a whole number we can multiply by 3 to get 20? No. So 3 is not a factor of 20.
Can we multiply start color #bc2612, 4, end color #bc2612 by a whole number to get 20? Yes. start color #bc2612, 4, end color #bc2612, times, start color #bc2612, 5, end color #bc2612, equals, 20. So start color #bc2612, 4, end color #bc2612 and start color #bc2612, 5, end color #bc2612 are a factor pair.
start color #0c7f99, 1, end color #0c7f99start color #208170, 2, end color #208170start color #bc2612, 4, end color #bc2612start color #bc2612, 5, end color #bc2612start color #208170, 10, end color #208170start color #0c7f99, 20, end color #0c7f99
The next counting number is 5. Since 5 already appears on the list, we now have found all of the factor pairs for 20.
Match the factor pairs for 40.

## Multiples

Multiples are numbers that result when we multiply one whole number by another whole number. The first four multiples of start color #6495ed, 3, end color #6495ed are 3, comma, 6, comma, 9, and 12 because:
start color #6495ed, 3, end color #6495ed, times, 1, equals, 3
start color #6495ed, 3, end color #6495ed, times, 2, equals, 6
start color #6495ed, 3, end color #6495ed, times, 3, equals, 9
start color #6495ed, 3, end color #6495ed, times, 4, equals, 12
Some other multiples of start color #6495ed, 3, end color #6495ed are 15, comma, 30 and 300.
start color #6495ed, 3, end color #6495ed, times, 5, equals, 15
start color #6495ed, 3, end color #6495ed, times, 10, equals, 30
start color #6495ed, 3, end color #6495ed, times, 100, equals, 300
We can never list all of the multiples of a number. In our example, 3 could be multiplied by an infinite number of numbers to find new multiples.

### Practice problems

The first multiple of any number is the number itself.
7, times, 1, equals, 7.
What are the next two multiples of 7?
7, times, 2, equals

7, times, 3, equals

The list shows multiples of 4.
4, comma, 8, comma, 12, comma, 16, comma, point, point, point
What is the next multiple of 4?

The list shows multiples of 8.
Fill in the missing multiples.
8, comma, 16,
, 32, comma, 40, comma, 48,
...

Which of the following numbers are multiples of 6?

### Picturing multiples

The following pictures show multiples of 4.
4, times, 1, equals, 4
An array of 4 ladybugs arranged in 1 row of 4 ladybugs.
4, times, 2, equals, 8
An array of 8 ladybugs arranged in 2 rows of 4 ladybugs each.
4, times, 3, equals, 12
An array of 12 ladybugs arranged in 3 rows of 4 ladybugs each.
The next box will include the next multiple of 4.
How many ladybugs will be in the next box?

### How do factors and multiples relate?

start color #e07d10, 4, end color #e07d10 and start color #1fab54, 7, end color #1fab54 are both factors of start color #11accd, 28, end color #11accd because they both divide evenly into start color #11accd, 28, end color #11accd.
start color #11accd, 28, end color #11accd is a multiple of start color #e07d10, 4, end color #e07d10, comma and it is also a multiple of start color #1fab54, 7, end color #1fab54.
Use the numbers 32 and 4 to complete the following sentences.
is a factor of
.
is a multiple of
.

## Practice with factors and multiples

Which of the following are factors of 10?

Which of the following are multiples of 10?

We know that 9, times, 6, equals, 54
So, which of the following statements are also true?

## Factors and multiples challenge

Factors and multiples are used when solving problems about the side lengths and areas of rectangles.
A rectangle has an area of 50 square cm.
Which of the following could be the side lengths of the rectangle?