TOPIC: WHOLE NUMBERS AND DECIMAL NUMBERS

A Prime Number can be divided evenly only by 1, or itself. And it must be a whole

number greater than 1.

Example: 5 can only be divided evenly by 1 or 5, so it is a prime number.

But 6 can be divided evenly by 1, 2, 3 and 6 so it is NOT a prime number (it is a

composite number).

Here is a table of all Prime Numbers up to 200:

Example 1:

a. Write down all the factors of 24

b. State which of these factors are prime numbers

c. Express 24 as a product of its factors.

Solution:

a. Factors of 24 are: 1, 2, 3, 4, 6, 8, 12, 24.

b. Prime factors of 24 are: 2 and 3

c. 24 = 2 x 2 x 2 x 3 = 23 x 3

Notice that 2 x 2 x 2 = 23 is in index form.

Example 2: Express 104 as a product of its prime factors in index form.

Solution: Method 1 : divide 104 by the prime number 2,3,4,7 in turn until it will

not divide further.

2 104

2 52

2 26

13 13

1

104 = 2 x 2 x 2 x 3 = 23 x 13

Factors of 104 = 1, 2, 4, 8, 13, 26, 52 and 104.

2 3 5 7 11 13 17 19 23

29 31 37 41 43 47 53 59 61 67

71 73 79 83 89 97 101 103 107 109

113 127 131 137 139 149 151 157 163 167

173 179 181 191 193 197 199

LEAST COMMON MULTIPLES (LCM)

Example 3: find the LCM of 22, 30 and 40.

Solution:

Method 1: express each number as a product of its Prime Factors

22 = 2 x 11

30 = 2 x 3 x 5

40 = 2 x 2 x 2 x 5

The prime factors of 22, 30 and 40 are: 2, 3, 5 and 11. The highest power of each

prime factor must be in the LCM.

These are 23, 3, 5 and 11.

Thus, LCM = 23 x 3 x 5 x 11

= 8 x 3 x 5 x 11

= 1,320

Method 2: divide 22, 30 and 40 by prime numbers in turn until it will not divide

further

2 22, 30, 40

2 11, 15, 20

2 11, 15, 10

3 11, 15, 5

5 11, 5, 5

11 11, 1, 1

1, 1, 1

LCM = 2 x 2 x 2 x 3 x 5 x 11

= 23 x 3 x 5 x 11

= 8 x 3 x 5 x 11

= 1,320

HIGHEST COMMON FACTORS (HCF)

Example 4: Find the HCF of 504 and 588.

Solution: Express each number as a product of its Prime Factors

2 504 2 588

2 252 2 294

2 126 3 147

3 63 7 49

3 21 7 7

7 7 1

1

504 = 2 x 2 x 2 x 3 x 3 x 7

588 = 2 x2 x 3 x7 x 7

HCF is the product of the common Prime Factors

HCF = 2 x 2 x 3 x 7

= 84

Example 5: Find the HCF of 72, 108, 54

Solution:

2 72 2 108 2 54

2 36 2 54 3 27

2 18 3 27 3 9

3 9 3 9 3 3

3 3 3 3 1

1 1

72 = 2 x 2 x 2 x 3 x 3

108 = 2 x 2 x 3 x 3 x 3

54 = 2 x 3 x 3 x 3

HCF = 2 x 3 x 3

=18

TOPIC: SQUARE AND SQUARE ROOT OF NUMBERS

Perfect Squares: A perfect square is the result got when a whole number is

multiply by itself e.g.

1 x 1 = 1

2 x 2 = 4

4 x 4 =16

9 x 9 = 81

1, 2, 4 and 81 are all perfect squares. They are simply called squares.

4 is a perfect square and it is square root of 16.

Example 1: Fin the square of (i) 20 and (ii) 101

Solution: (i) 20 x 20 = 400

(ii) 101 x 101 = 10,201

Example 2: Find the smaller number by which 50 will be multiplied so that the

product is a Perfect Square.

Solution:

2 50

5 25

5 5

1 50 = 2 x 52

The index of 2 here is 1. It must be eve i.e. 2 or 4 therefore; there is need for at

least one more 2 to make the index of 2 even. The least number is 2.

50 x 2 = 2 x 2 x 52

100 = 22 x 52

The required number is 2

Example 2: Find the smaller number by which 162 will be multiplied so that the

product is a Perfect Square.

Solution:

162 = 2 x 81

= 2 x 9 x 9

= 2 x 3 x 3 x 3 x 3

= 2 x 34

162 x 2 = 2 x 2 x 34

324 = 22 x 43

SQUARE ROOTS OF PERFECT SQUARES

Example 5: Find the square root of 7,056

Solution:

7056 = 2 x 3528

= 2 x 2 x 1764

= 2 x 2 x 2 x 882

= 2 x 2 x 2 x 2 x 441

= 2 x 2 x 2 x 2 x 3 x 147

= 2 x 2 x 2 x 2 x 3 x 3 x 49

= 2 x 2 x 2 x 2 x 3 x 3 x 7 x 7

= 22 x 22 x 33 x 77

= 2 x 2 x 3 x 7

= 84.

Example 6: Find the square root of the following:

Solution:

a.

b.

c.