# Lesson 6 – Common And Least Common Multiples. (L. C. M)

Objectives:

At the end of the lesson, students should be able to find the least common multiples of a given set of numbers.

The common multiples (C.M) of any set of numbers are the numbers that can be found in the multiples of both numbers.

The least common multiple (L.C.M) is the smallest number that is a multiple of two or more given numbers. It is also referred to as the lowest common multiple or the smallest common multiple.

There are several ways of finding the L.C.M of a set of numbers.

LISTING METHOD.

To find the L.C.M of any pair of numbers using this method, follow these steps:

1. List the multiples of the given numbers.
2. Identify and list the common multiples
3. Pick out the smallest or lowest number in the list. This is the lowest common multiple.

Example: 1

Find the common multiples of these pairs of numbers.

1. 2 and 3
2. 4 and 5
3. 6 and 9
4. 4, 6 and 9

Solution

1. List the first 12 multiples of 2 and 32 = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24}

3 = {3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36}

List all the common factors.

C.M of 2 and 3 = {6, 12, 18 and 24}

2. 4 and 5
3. 4 = {4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48}

5 = {5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 16}

C.M = {20 and 40}

4. 6 and 9
5. 6 = {6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72}

9 = {9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108}

C.M = {18, 36, 54, 72}

6. 4, 6 and 9

4 = {4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 54}

6 = {6, 12, 18, 24, 36, 45, 54, 63, 72, 81, 90, 99, 108}

9 = {9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108}

C.M = {36)

Note that in examples “d”, the only common multiple of 4, 6 and 9 is {36} if you list only the first 12 multiples of the numbers. However, if you go ahead beyond 12 multiples, you will find that there are definitely more common multiples.

Example: 2

Find the least common multiple of each of the following pairs of numbers.

1. 5 and 10      d)  3, 4 and 6
2. 6 and 5      e)  6, 9 and 12
3. 4 and 7

Solutions

1. 5 and 10

5 = {5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60}

10 = {10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 120}

C.M = {10, 20, 30, 40, 50, 60}

L.C.M = 10

1. 6 and 5

6 = {6, 12, 18, 24, 30, 42, 48, 54, 60, 66, 72}

5 = {5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60}

CM = {30, 60}

LCM = 10

1. 4 and 7

4 = {4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48}

7 = {7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84}

CM = {28}

LCM = 28

1. 3, 4 and 6

3 = {3, 6, 9, 12, 15, 18, 21. 24, 27, 30, 33, 36}

4 = {4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48}

6 = {6, 12, 18, 24, 30, 36, 42, 48, 5, 60, 66, 72}

CM = {12, 24, 36}

LCM = 12

1. 6, 9 and 12

6 = {6, 12, 18, 24, 30, 36, 42, 48, 5, 60, 66, 72}

9 = {9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108}

12 = {12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 132, 144}

CM = {36, 72}

LCM = 36

Unit 8 Lesson 6: Exercise 1

List the common multiples of the following sets of numbers.

1. 6 and 9
2. 9 and 12
3. 15 and 20
4. 7 and 14
5. 5, 4 and 12
6. 4, 10 and 15
7. 9, 10 and 15
8. 4, 9 and 36
9. 8, 9 and 12
10. 8, 10 and 40

1) 6 and 9
6 = {6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72}
9 = {9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108}
CM = {18, 36, 54, 72}2) 9 and 12
9 = {9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108}
12 = {12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 144}
CM = {36, 72, 108}

3) 15 and 20
15 = {15, 30, 45, 60, 75, 90, 105, 120, 135, 150, 165, 180}
20 = {20, 40, 60, 80, 100, 120, 140, 160, 180, 200, 220, 240}
CM = {60, 120, 180}

4) 7 and 14
7 = {7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84}
14 = {14, 28, 42, 56, 70, 84, 98, 112, 126, 140, 154, 168}
CM = {14, 28, 42, 56, 70, 84}

5) 5, 4 and 12
5 = {5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60}
4 = {4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 60}
12 = {12, 24, 36, 48, 60, 72, 84, 96, 120, 132, 144}
CM = {60}

6) 4, 10 and 15
4 = {4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 60}
10 = {10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120}
15 = {15, 30, 45, 60, 75, 90, 105, 120, 135, 150, 165, 180}
CM = {60}

7) 9, 10 and 15
9 = {9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108}
10 = {10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120}
15 = {15, 30, 45, 60, 75, 90, 105, 120, 135, 150, 165, 180}
CM = {90}

8) 4, 9 and 36
4 = {4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 60}
9 = {9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108}
36 = {36, 72, 108, 144, 180, 216, 252, 288, 324, 360, 396, 432}
CM = {3}

9) 8, 9 and 12
8 = {8, 16, 24, 32, 40, 48, 56, 72, 80, 88, 96}
9 = {9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108}
12 = {12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 132, 144}
CM = {72}

10) 8, 10 and 40
8 = {8, 16 24 32, 40, 48, 56, 72, 80, 88, 96}
10 = {10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120}
40 = {40, 80, 120, 160, 200, 280, 320, 360, 400, 440, 480}
CM = {40, 80}

Unit 8 Lesson 6: Exercise 2

Find the least common multiples of the following sets of numbers.

1. 3 and 7
2. 6 and 8
3. 7 and 9
4. 8 and 4
5. 5 and 10
6. 3 and 5
7. 6 and 9
8. 8 and 12
9. 15 and 30
10. 25 and 50

1) 3 and 7
3 = {3, 6, 9, 12, 15, 18, 21. 24, 27, 30, 33, 36}
7 = {7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84}
CM = {21}
LCM = 212) 6 and 8
6 = {6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72}
8 = {8, 16 24 32, 40, 48, 56, 72, 80, 88, 96}
CM = {24, 48}
LCM = 24

3) 7 and 9
7 = {7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84}
9 = {9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108}
CM = {63}
LCM = 63

4) 4 and 8
4 = {4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 60}
8 = {8, 16 24 32, 40, 48, 56, 72, 80, 88, 96}
CM = {8, 16, 24, 32, 40, 48}
LCM = 8

5) 5 and 10
5 = {5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60}
10 = {10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120}
CM = {10, 20, 30, 40, 50, 60}
LCM = 10

6) 3 and 5
3 = {3, 6, 9, 12, 15, 18, 21. 24, 27, 30, 33, 36}
5 = {5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60}
CM = {15, 30}
LCM = 15

7) 6 and 9
6 = {6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72}
9 = {9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108}
CM = {18, 36, 54, 72}
LCM = 18

8) 8 and 12
8 = {8, 16 24 32, 40, 48, 56, 72, 80, 88, 96}
12 = {12, 24, 36, 48, 60, 72, 84, 96, 1088, 120, 132, 144}
CM = {24, 48, 72, 96}
LCM = 24

9) 15 and 30
15 = {15, 30, 45, 60, 75, 90, 105, 120, 135, 150, 165, 180}
30 = {30, 60, 90, 120, 150, 180, 210, 240, 270, 300, 330, 360}
CM = 30, 60, 90, 120, 150, 180}
LCM = 30

10) 25 and 50
25 = {25, 50, 75, 100, 125, 150, 175, 200, 225, 250, 275, 300}
50 = {50, 100, 150, 200, 250, 300, 350, 400, 450, 600}
CM = {50, 100, 150, 200, 250, 300}
LCM = 50

Unit 8 Lesson 6: Exercise 3

Write the least common multiple of each of the following sets of numbers.

1. 3 and 6
2. 3 and 9
3. 6 and 12
4. 4 and 6
5. 8 and 9
6. 4, 6 and 8
7. 3, 5 and 10
8. 3, 5 and 15
9. 30, 50 and 15
10. 25, 30 and 35

1) 3 and 6
3 = {3, 6, 9, 12, 15, 18, 21. 24, 27, 30, 33, 36}
6 = {6, 12, 18, 24, 30, 36, 42, 48, 5, 60, 66, 72}
CM = {6, 12, 18, 24, 30, 36}
LCM = 62) 3 and 9
3 = {3, 6, 9, 12, 15, 18, 21. 24, 27, 30, 33, 36}
9 = {9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108}
CM = 9, 18, 27, 36}
LCM = 9

3) 6 and 12
6 = {6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72}
12 = {12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 132, 144}
CM = {12, 24, 36, 48, 60, 72}
LCM = 12

4) 4 and 6
4 = {4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48}
6 = {6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72}
CM = {12, 24, 36, 48}
LCM = 12

5) 8 and 9
8 = {8, 16 24 32, 40, 48, 56, 72, 80, 88, 96}
9 = {9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108}
CM = {72}
LCM = 72

6) 4, 6 and 9
4 = {4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48}
6 = {6, 12, 18, 24, 30, 36, 42, 48, 5, 60, 66, 72}
9 = {9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108}
CM = {36}
LCM = 36

7) 3, 5 and 10
3 = {3, 6, 9, 12, 15, 18, 21. 24, 27, 30, 33, 36}
5 = {5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60}
10 = {10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120}
CM = {30}
LCM = 30

8) 3, 5 and 15
3 = {3, 6, 9, 12, 15, 18, 21. 24, 27, 30, 33, 36}
5 = {5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60}
15 = {15, 30, 45, 60, 75, 90, 105, 120, 135, 150, 165, 180}
CM = {15, 30}
LCM = 15

9) 30, 50 and 15
30 = {30, 60, 90, 120, 150, 180, 210, 240, 270, 300, 330, 360}
50 = {50, 100, 150, 200, 250, 300, 350, 400, 450, 500, 550, 600, 650}
15 = {15, 30, 45, 60, 75, 90, 105, 120, 135, 150, 165, 180}
CM = {150}
LCM = 150

10) 25, 30 and 60
25 = {25, 50, 75, 100, 125, 150, 175, 200, 225, 250, 275, 300}
30 = {30, 60, 90, 120, 150, 180, 210, 240, 270, 300, 330, 360}
60 = {60, 120, 180, 210, 240, 300, 360, 420, 480, 540, 600, 660, 720}
CM = {300}
LCM = 300

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