# Lesson Notes By Weeks and Term - Junior Secondary School 1

LEAST COMMON MULTIPLE AND HIGHEST COMMON FACTOR OF WHOLE NUMBERS

SUBJECT: MATHEMATICS

CLASS:  JSS 1

DATE:

TERM: 1st TERM

WEEK THREE

TOPIC: LEAST COMMON MULTIPLE AND HIGHEST COMMON FACTOR OF WHOLE NUMBERS

CONTENT

• Multiples
• Common Multiples
• Least Common Multiples (LCM)
• Highest Common Factors (HCF).

Multiples

Multiples mean finding the product of a positive integer with another positive integer. Simply put, when a whole number multiples another whole number, the result obtained is called the multiple of either of those numbers.

The first four multiples of 12 are

12 x 1    = 12

12 x 2    = 24

12 x 3    = 36

12 x 4 =48

Thus, we write 12, 24, 36 and 48 as the first for multiples of 12.

Note: Every whole number has an infinite number of multiples

Every whole number has a finite number of factors.

Example 1

Find the next five multiple of the following whole numbers.

(a) 4        (b) 8        (c ) 11.

Solution

In these questions, the numbers are not to be included because it reads next.

(a) 4,    4 x 1     =  4    (not included)

4 x 2    =   8

4 x 3    =   12

4 x 4    =   16

4 x 5    =   20

4 x 6    =   24

:. The next five multiples of 4 are 8, 12, 16, 20 and 24.

(b) 8,   8 x 1     =8        multiple of 11

8 x 2    =16        11 x 1    =11

8 x 3    =24        11 x 2    = 22

8 x 4    =32        11 x 3    = 33

8 x 5    =40        11 x 4   = 44

8 x 6    = 48        11 x 5    = 55

11 x 6    = 66

the next five multiples of         the next five multiples of 11 are  22, 33,44,55 and 66

8 are 16, 24, 32, 40 and 48.

Example 2

Which of the following numbers 18, 20, 27 36 and 50 are

1. multiples of 2
2. multiples of 3
3. multiples of 4.

Solution

When a number can be divided exactly by another number it means the quotient is a multiple of the divisor. 18, 20, 27 36, 50

1. multiples of 2 are 18, 20, 27 36, 50
2. multiples of 3 are 18,  27 and  36
3. multiple of 4 are  20 and  36.

EVALUATION

1. 1.Find the next 7 multiples of the following numbers.

(a) 15        (b) 25      (c ) 13.

1. Which of the following whole numbers 37, 68, 51, 128, 85 and 187 are

(a) multiples of 2

(b) multiples of 3

(c) multiples of 5

(d) multiples of 17.

Common multiples

When two or more numbers have a multiple in common, then the numbers is known as a common multiple.

Example

Find the first two common multiples of 4.6 and 8.

Solution

Their multiple are as shown below;

4 = 4, 8, 12, 16, 20 (24) 28,32, 36, 40, 44, (48) 52,56,60,64……

6 = 6, 12, 18,(24), 30, 36, 42, (48), 54, 60, 66, 72.

8= 8, 16,(24), 32, 40, (48), 56, 64, 72,

Considering the three whole numbers, their first two common multiples are

24 and 48.

Examples

Write down three common multiples of the following sets of numbers

(a) 5 and 6

(b) 3, 10 and 15.

Solution

(a) First three common multiples of 5 and 6.

5 = 5, 10, 15, 20, 25, (30),35, 40 45, 50, 55 (60) 65, 70,75,80 85,(90) 95,100, 105,110,115,120.

6= 6, 12,18,24, (30),36, 42,48,54,(60),66,72,78,84, (90),96,102,106,114,120.

:.The first three common multiples of 5 and 6 are, 30 60 and 90.

(b) First three common multiples of 3, 10 and 15

3= 3,6,9,12,15,18,21, 24,27, (30),33,36,39,42,45,81,84,87,(90),93,96.

10= 10, 20, (30), 40, 50, (60) 70, 80 (90),100,110, etc

15 = 15, (30), 45,(60), 75, (90), 105, 120.

:.  The first three common multiples of 3, 10, and 15 are  30, 60 and 90.

EVALUATION

Find the first  four common multiples of the following sets of numbers

(a) 4 and 7        (b) 2,5 and 7    (c ) 3, 6, 9.

1. Essential Mathematic for JSS1 by AJS Oluwasanmipg 32
2. New General Mathematics for JSS1 by M.F Macraeetalpg 26-27.

Least Common Multiples (LCM)

You can find the least common multiples of two or more numbers by listing as many multiples as you need until you have one that is common to both or all the numbers.

For instance to find the LCM of 24 and 15

Multiples of 24 = 24, 48, 72,96, 120………

Multiples of 15 = 15, 30, 45, 60, 75,90, 105, 120.

Although the numbers will have many common multiples but, looking at what we are after, that is the least of the common multiples, the answer will be 120.

:. LCM of 24 and 15 = 120.

Rather than writing out a long list of multiples for each number, you can use the prime factors method to find the LCM.  This is the method we are going to apply.

Example 1

Find the LCM of the following whole numbers:

(a) 24 and 15        (b) 8 and 45    ( c ) 16 and 18   (d) 90, 105 and 210.

Solution

(a) The LCM of 24 and 15

2     24 15        2   8   45

2     12  15                    2   4   45

2     6    15                                                   2   2   45

3     2     5                                                        3   1   45

5     1     1                                                        3   1   15

3   1    5

5   1    1

LCM = 2 x 2 x 2 x 3 x 5= 120            Lcm = 2 x 2 x 2 x 3 x 3 x 5  = 360

:.LCM of 24 and 15 = 120                ;. LCM of 8 and 45 = 360.

(c) LCM of 16 and 18                (d) LCM of 90, 105 and 210

2     16  18                 2   90  105  210

2      8    9                      3   45  105  105

2    4    9                          3   15   35    35

2      2    9                              5    5    35    35

2      1    9                              7    1     7      7

3      1    3                                  1     1      1

3      1    1cc

LCM = 2 x 2 x 2 x  x 2 x3 x3 = 144        LCM = 2 x 3 x 3 x 5 x 7 = 630

:. LCM of 16 and 18 = 144        :. LCM of 90, 105 and 210 is = 630.

Given that the numbers are expressed as a product of prime factors, the lcmis the product of the prime factors of the numbers without double counting.

Example 2

Find the LCM of the following .Leave your answers in prime factors.

(a) 2 x 2 x 3,        (b )2 x 2 x 5

2 x 2 x 2 x 5                      3 x 5 x 7

2 x 2 x5              2 x 3 x 3 x 3

2x 2 x 3 x 3 x 5    3 x 5 x 5 x 7.

Solution

(a) 2 x 2 x 3

2 x 2 x 2 x 5

2 x   2   x 5

2 x  2 x 3 x 3 x 5

LCM = 2 x 2 x2 x 3 x 3 x 5.

(b) 2 x 3 x 3

3x 5x 7

2 x 3 x 3 x 3

3 x 5 x 5 x 7

LCM = 2 x 3 x 3 x 3 x 5 x 5 x 7.

EVALUATION

1.Find the LCM of the following

(a) 4,6, and 9        (b) 6, 8, 10 and 12    (c ) 9, 10,12 and  15  (d)   108 and 360.

1. Find the  Lcm of the following leaving your answers in index form.

(a) 2 x 2 x 2 x 3 x 3            (b) 3  x 3 x 5

2 x 3 x 5 x5                    2 x 3 x 7

2 x 2 x 3 x 3 x 5                2 x 5 x 5 x 7

(c)  23 x 32 x 5

3 x 53  x 72

24  x 3 x 72,

32  x 52  x 73

Highest Common Factor

Highest common factor (HCF) of two or more numbers is the largest number that divides exactly into all the numbers.

Example 1

Find the HCF of 21 and 84.

Solution

3    21                                                      2     84

7    7                                                    2   42

1                                                    3     21

7    7

1

21 = 3 x 7

84  = 2 x 2 x 3 x 7

HCF = 3 x 7  = 21

Example 2

Find the HCF of 195 and 330.

Solution

3      195

5      65

1. 13

11

HCF of 195 and 330

195 =  3 x 5 x 13

330 =  2 x 3x5x11

HCF = 3 x 5 = 15

1. 330
2. 165

5       55

11     11

1

Example 3

Find the HCF of 288, 180 and 106. leave your answer in index form.

Solution

2      288                     2          180                   2           108

2      144                    2            90                   2           54

2        72                    3            45                   3            27

2       36                     3           15                   3             9

2        18                    5            5                    3             3

3        9                                    1                                   1

3        3

1

288 = 2 x 2 x 2 x2x2 x 3 x 3

180 =  2x2 x 3 x 3

108 = 2x2 x3 x  3x3

HCF =    2x 2x 3 x 3 = 22 x 32 …….index form

= 36 (ordinary form).

Example 4

Find the HCF of the following . Leave the answers in prime factors and use index notation.

(a) 23 x 32 x 7

22 x 3 x 52

22 x 33 x 5

(b) 23  x 52 x 7

22 x 32  x 5

33 x 53  x 72

Solution

(a)23 x 32 x 7  = 2 x 22 x 3 x 3 x7

22 x 3 x 52   =  22 x 3 x 52

22 x 33 x 5  =  22x 3 x 32 x 5

HCF = 22 x 3  index form

4 x 3 = 12.

(b) 23  x 52 x 7  = 23x  52 x 7

22 x 32  x 5 = 22 x 32x 5

33 x 53  x 72= 33 x 53 x 72

The factor that is common is in 5

:. HCF = 5

EVALUATION

1.Findthe HCF of the following;

1. leaving your answers in index form
2. leaving your answers in whole number

(a) 160, 96 and 224

(b) 189, 279and 108

(c) 126, 234 and 90.

1. Find the HCF of the following .Leave your answers in prime factors and use index  notation.

22x 33 x 5

21 x 34 x 5

2 x 35 x 72

(b)   23 x 33 x 53

24 x 3 x 52 x 7

25 x 32 x 5 x 72

1. New  General Mathematics for jss 1 by m.Fmacraeetalpg 25-26
2. Essential Mathematics for jss1 by AjSOluwasanmi, pg 31.

WEEKEND ASSIGNMENT

1. The value of  23 x 32 is (a) 1291 (b) 658   (c) 729   (d)7 36   (e) 54
2. The LCM of 12 and 15 is (a) 90 (b) 60    (c) 30 (d) 120   (e) 180
3. The HCF of 63and 90 is (a) 7 (b) 3        (c) 12   (d) 6   (e) 9
4. The first three common multiples of 3 and 11 are (a) 3, 33, 66,    (b) 11, 33, 66    ( c ) 33, 66, 99   (d) 33, 44, 55   (e) 33, 22, 11.
5. Which of the following whole numbers 22, 11, 54, 35, 40, 75, and 105 is /are multiples of 5? (a) 11, 22, 35     (b) 35, 40, 75, 105  ( c ) 54, 35, 40, 75, 105 (d) 35, 54, 40, 75  (e) 105,75,40,35,54.

THEORY

1. Give the first five multiples of the following
2. 5     II 7    III. 11

B Write down four common multiples of the following sets of numbers

1. 3, 4 and 5     II. 3, 10 and 15.

2a. Find the LCM of

1. 9, 24, 32, and 90     II. 23 x 32 x 5 x 7

3 x 53 x 72

24 x 3 x 72

32 x 52 x 73

1. Find the HCF of
2. 126, 234 and 90    ii. 23 x 33 x 53
3. 63, 42, and 21 24 x 3 x 52 x 7

25 x 32 x 5 x 72