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:
Example: 1
Find the common multiples of these pairs of numbers.
Solution
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}
5 = {5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 16}
C.M = {20 and 40}
9 = {9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108}
C.M = {18, 36, 54, 72}
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.
Solutions
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
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
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
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
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.
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.
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.
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