Lowest Common Multiple (LCM) & Highest Common Factor(HCF) JSS1 Mathematics Lesson Note

COMMON MULTIPLES AND FACTORS

A prime number is a number that can only be divided by itself. It has two factors, which are 1 and itself. Examples of prime numbers are 3, 5, 7, 11, 13, 17, 19, etc.

Multiples: A multiple of a number is obtained by multiplying it by any whole number. For example, the multiples of 4 are 4, 8, 12, 16, 20, 24, etc.

Factors: The factor of a number is the whole number that divides the number exactly.

Example 1: (a) Find all the factors of 18

             (b) State which of these factors is even

             (c) State which of these factors are prime numbers

             (d) Write the first three multiples of 18

Solution

(a)     Factors of 18 are 1, 2, 3, 6, 9, and 18

(b)     The even numbers are 2, 6, and 18

(c)     The prime numbers are 2 and 3

Example 2: Find the factor pairs of 56

Solution:

1 × 56

2 × 28

4 × 14

7 × 8

Therefore, the factors of 56 are 1, 2, 4, 7, 8, 14, 28, and 56.

Product of a Prime Factor

A prime factor is a factor that is also a prime number. You can find the product of prime factors of a number using a prime factor tree method or using the method of repeatedly dividing by the prime numbers.

Example 2: Express the following numbers, 56 and 108, as products of prime factors in index form.

Solution:

Method 1: Dividing repeatedly by using prime numbers

2          56                                                                                       2          108

2          28                                                                                       2          54                                                                      

2          14                                                                                       3          27

7          7                                                                                          3          9

             1                          Index form = 23 x 733

index form = 22 x 32

Method 2: Factor tree 

                56                                                                       108

             2                          28                                       2                          54

                             2                          14                                       2                          27

                                             2                          7              

Note that the numbers must be prime numbers

EXAMPLE 1: Find the L C M of 18 and 24

Solution:

METHOD 1                                                                                    METHOD 2       

2          18       24                                                                               18 = 2 ×3 ×3

2          9          12                                                                               24 = 2 ×2 ×2 ×3

2          9          6                                                                                  L C M = 2 ×2 ×2 ×3 ×3

3          9          3                                                                                     = 72

3          3          1

             1          1

LCM = 2 × 2 × 2 × 3 × 3 = 72        

Example 2: Find the LCM of 72 and 90

Solution: 

2          72       90                                                       72 = 2 X X 2 X 3 X 3

2          36       45                                                       90 = 2 X 3 X 3 X 5

2          18       45                                                       L C M = 2 X 2 X 2  X 3 X 3 X 5

3          9          45                                                        = 360

Example 3: Find the HCF of  72 and 96

Solution: find the prime product of the number and pick the common ones

72 = 2 * 2 * 2 * 3 * 3

96 = 2 * 2 * 2 * 2 * 2 * 3

HCF = 2 * 2 * 2 * 3 = 24