Multiples

Multiples are numbers that can be divided evenly by other numbers. Multiples are also the product when two or more factors are multiplied.

Examples

A) 15 is a multiple of 3 and 5, because 3 × 5 = 15 B) 24 is a multiple of 8 and 3, because 8 × 3 = 24

Strategies for Determining Multiples Multiplication using mind math or pencil and paper Example

Identify the first five multiples of 7. Think of your multiplication facts. 7×1=7 7 × 2 = 14 7 × 3 = 21 7 × 4 = 28 7 × 5 = 35

The first five multiples of 7 are: 7, 14, 21, 28, 35

You can determine multiples of numbers by skip counting. Think of multiplication facts to help you. Count up to find the next three multiples of these numbers: 4

4, 8, 12, 16, 20, 24, 28

9

9, 18, 27, 36, 45, 54

12

12, 24, 36, 48, 60, 72, 84

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Addition using a calculator Example

Identify the first five multiples of 3. 1. Turn on your calculator.

3.

2. Type in the number 3. Then hit the 4. Now hit the

5. Hit the

=

+ 3

key. key (the multiple you are counting up by).

key and write down the number. Hit the

=

two more

times, and write down each number displayed in order. The first six multiples of 3 are: 3

6

9

12

15

Multiplication using a calculator Your calculator can also find multiples of numbers using multiplication.

Example

Find the first five multiples of 8. 1. Turn on your calculator.

8.

2. Type in the number 3. Then hit the 4. Now hit the 5. Hit the

=

× 1

key. key.

key and write down the number. 8×1=8

6. Repeat steps 1 to 5, but increase the value of the key in step 4 by 1 each time 8 8 8 8

× × × ×

2 = 16 3 = 24 4 = 32 5 = 40

The first five multiples of 8 are: 8

16

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24

32

40

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Sets of Multiples The multiples of a number are represented as sets, lists or groups of numbers.

Example

The first six multiples of 8 are 8, 16, 24, 32, 40, 48.

The set of the first six multiples of 8 is:

8: {8, 16, 24, 32, 40, 48} colon

Multiples of 8 in increasing order from left to right separated by commas.

The { } signs show that numbers inside belong to a set of numbers.

If a specific number of digits in a set is not asked for, write the set of multiples like this: The set of multiples of 6 is: 6: {6, 12, 18 …} Write the first 3 multiples in order.

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Write three periods to indicate that the pattern continues.

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Lowest Common Multiples Hamburger patties and buns are often sold in different quantities. To purchase exactly the same number of patties and buns, we could calculate the lowest common multiple of each package of buns and hamburger patties.

The Lowest Common Multiple (LCM) of a pair of numbers is the smallest multiple that is common to both the numbers.

Follow these steps to find the lowest common multiple of two numbers. 1. List multiples of both numbers in order from least to greatest. (You do not need to list all the possible multiples for both numbers. List a few at a time for each number until the same multiple appears in both lists.) 2. Determine the lowest multiple that is the same for both numbers.

Example

If hamburger patties were sold in packages of 6, and buns in bags of 8, how many would we need to buy to have the same number of patties and buns?

Find the lowest common multiple of 6 and 8. Multiples of 6 include: 6: {6, Multiples of 8 include: 8: {8,

12, 16,

18,

24,

24,

30…} The lowest common multiple of 6 and 8 is 24

32 …}

We would need to purchase 24 patties and 24 buns to have an equal number of both. Remember there are 6 patties in a package. 24 ÷ 6 = 4 24 ÷ 8 = 3 There are 8 buns in a bag. We would need 4 packages of hamburger patties and 3 packages of buns.

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Using Multiples to Solve Problems Examples A) The number 22 bus comes to James’ bus stop every 12 minutes. The number 27 arrives every 8 minutes. In how many minutes will both buses arrive at the stop together? Find the multiples of 12 and 8. Multiples of 12 are 12, 24, 36, etc. Multiples of 8 are 8, 16, 24, etc. Conclusion: Both buses will arrive at James’ stop together every 24 minutes.

B) Bonnie rotates the tires of her truck every 8000 km and changes the oil every 5000 km. She will rotate the tires and change the oil at the same time after how many kilometres? Find the multiples of 8000 and 5000. (Hint: Use 8 and 5) Multiples of 8 are 8, 16, 24, 32, 40, etc. Multiples of 5 are 5, 10, 15, 20, 25, 30, 35, 40, etc. Conclusion: Bonnie will rotate her tires and change her oil together every 40 000 km.

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Practice: Finding Multiples and Sets of Multiples

1. Use your calculator or mind math to find the missing multiples. Then, write each group of multiples as a set. 4

12

6 100

2. Fill in the missing numbers and then answer each question.

1

2

3

4

6

8

This code was cracked by counting by multiples of______.

2

4

6

10

12

16

This code was cracked by counting by multiples of______.

5

10

20

25

35

This code was cracked by counting by multiples of______.

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10

20

30

50

60

80

This code was cracked by counting by multiples of______.

20

40

60

80

120

140

This code was cracked by counting by multiples of______.

3. The first three multiples of 6 are listed below. Fill in the next three and then finish writing the multiples as a set by filling in the blanks. 6×1= 6 6 × 2 = 12 6 × 3 = 18 6×4= 6×5= 6×6=

6 : { 6, 12, 18,

,

,

}

4. The first three multiples of 12 are listed below. Fill in the next three and then finish writing the multiples as a set by filling in the blanks. 12 × 1 = 12 12 × 2 = 24 12 × 3 = 36 12 × 4 = 12 × 5 = 12 × 6 =

12 : { 12, 24, 36,

,

,

}

5. Is 72 a multiple of 9? Show your response.

6. Identify the 2 sets of numbers (all less than 10) that have a lowest common multiple of 18.

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7. Identify the 2 numbers (less than 10) that have a lowest common multiple of: a) 12

E.g.

6

and

2

4

and

3

b) 20 c) 15 d) 32

8. The school band needs refreshments and snacks for all 58 band members who will be rehearsing Saturday afternoon. Pop comes in cases of 12 and muffins come in packages of 6. Find the lowest common multiple of 12 and 6 that will give all 58 students a pop and muffin on Saturday.

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