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Course: 6th grade foundations (Eureka Math/EngageNY) > Unit 2

Lesson 4: Topic D: Foundations
Math
6th grade foundations (Eureka Math/EngageNY)
Module 2: Foundations
Topic D: Foundations
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Factors and multiples

Google Classroom
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 1 row with 12 dots.
1×12=12
An array of 12 circles arranged in 1 row of 12 circles.
12 dots can also be arranged in 2 rows with 6 dots per row.
2×6=12
An array of 12 circles arranged in 2 rows of 6 circles each.
Or, we can arrange 12 dots in 3 rows with 4 dots in each row.
3×4=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.
1, 12, 2, 6, 3, and 4 are all factors of 12.
We can make 12 with a row of 5 and a row of 7. So are 5 and 7 factors of 12?
An array of 12 circles arranges in 1 row of 5 circles and 1 row of 7 circles.
No. 5 and 7 are not factors because the dots are not divided into equal sized groups.
Which of the following arrangements are possible for 18 dots?
Choose all answers that apply:

So what are the factors of 18?
Choose all answers that apply:

Finding factors without pictures

We can find the factors for 16 without drawing dots by thinking about numbers that will divide into 16 evenly.
1 is a factor of 16 because 1 can be divided into 16 with no remainder.
16÷1=16
The quotient, which is 16, is also a factor of 16.
2 is a factor of 16 because 2 can be divided into 16 with no remainder.
16÷2=8
The quotient, which is 8, is also a factor of 16.
4 is a factor of 16 because 4 can be divided into 16 with no remainder.
16÷4=4
In this case the quotient is 4, which we have already discovered is a factor of 16.
The factors of 16 are 1,16, 2,8, and 4.
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
Not a factor
1
2
3
5
7
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 5,787.
Every number has itself as a factor.
41 is a factor of 41.
128 is a factor of 128.
4,379 is a factor of 4,379.

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 1 × 8 and 2 × 4. So the factor pairs for 8 are 1 and 8 and 2 and 4.
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 4 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 1 because we know that 1 is a factor for every number. We multiply 1×20, to get 20, so 20 is also a factor. We can list these factors as the outside ends of a list, leaving room in the middle for additional factors.
120
Now we check to see whether the next counting number, 2, is a factor.
Is there a whole number we can multiply by 2 to get 20? Yes. 2×10=20. So 2 and 10 are another factor pair.
121020
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 4 by a whole number to get 20? Yes. 4×5=20. So 4 and 5 are a factor pair.
12451020
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.
1

Multiples

Multiples are numbers that result when we multiply one whole number by another whole number. The first four multiples of 3 are 3,6,9, and 12 because:
3×1=3
3×2=6
3×3=9
3×4=12
Some other multiples of 3 are 15,30 and 300.
3×5=15
3×10=30
3×100=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.
7×1=7.
What are the next two multiples of 7?
7×2=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

7×3=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

The list shows multiples of 4.
4,8,12,16,…
What is the next multiple of 4?
Choose 1 answer:

The list shows multiples of 8.
Fill in the missing multiples.
8,16,
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
, 32,40,48,
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
...

Which of the following numbers are multiples of 6?
Choose all answers that apply:

Picturing multiples

The following pictures show multiples of 4.
4×1=4
An array of 4 ladybugs arranged in 1 row of 4 ladybugs.
4×2=8
An array of 8 ladybugs arranged in 2 rows of 4 ladybugs each.
4×3=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?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
ladybugs

How do factors and multiples relate?

4 and 7 are both factors of 28 because they both divide evenly into 28.
28 is a multiple of 4, and it is also a multiple of 7.
Use the numbers 32 and 4 to complete the following sentences.
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
is a factor of
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
.
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
is a multiple of
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
.

Practice with factors and multiples

Which of the following are factors of 10?
Choose all answers that apply:

Which of the following are multiples of 10?
Choose all answers that apply:

We know that 9×6=54
So, which of the following statements are also true?
Choose all answers that apply:

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?
Choose all answers that apply:

Mr. Trimble is putting out 36 chocolate chip cookies for the students in his art club.
If he arranges the cookies in 3 rows, there will be
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
cookies in each row.
If he arranges the cookies in
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
rows, there will be 4 cookies in each row.

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