1. Real Numbers

2. Real Numbers
1. Prime Factors
Every natural number can be written as a product of its prime factors.

3. Real Numbers
1. Prime Factors
Every natural number can be written as a product of its prime factors.
e.g. 324

4. Real Numbers
1. Prime Factors
Every natural number can be written as a product of its prime factors.
e.g. 324  4  81

5. Real Numbers
1. Prime Factors
Every natural number can be written as a product of its prime factors.
e.g. 324  4  81
 22  34

6. Real Numbers
1. Prime Factors
Every natural number can be written as a product of its prime factors.
e.g. 324  4  81
 22  34
2. Highest Common Factor (HCF)
1) Write both numbers in terms of its prime factors

7. Real Numbers
1. Prime Factors
Every natural number can be written as a product of its prime factors.
e.g. 324  4  81
 22  34
2. Highest Common Factor (HCF)
1) Write both numbers in terms of its prime factors
2) Take out the common factors

8. Real Numbers
1. Prime Factors
Every natural number can be written as a product of its prime factors.
e.g. 324  4  81
 22  34
2. Highest Common Factor (HCF)
1) Write both numbers in terms of its prime factors
2) Take out the common factors
e.g. 1176 and 252

9. Real Numbers
1. Prime Factors
Every natural number can be written as a product of its prime factors.
e.g. 324  4  81
 22  34
2. Highest Common Factor (HCF)
1) Write both numbers in terms of its prime factors
2) Take out the common factors
e.g. 1176 and 252
1176  6 196
 3  2  49  4
 3  23  7 2

10. Real Numbers
1. Prime Factors
Every natural number can be written as a product of its prime factors.
e.g. 324  4  81
 22  34
2. Highest Common Factor (HCF)
1) Write both numbers in terms of its prime factors
2) Take out the common factors
e.g. 1176 and 252
1176  6 196 252  4  63
 3  2  49  4  49 7
 3  23  7 2  22  32  7

11. Real Numbers
1. Prime Factors
Every natural number can be written as a product of its prime factors.
e.g. 324  4  81
 22  34
2. Highest Common Factor (HCF)
1) Write both numbers in terms of its prime factors
2) Take out the common factors
e.g. 1176 and 252
1176  6 196 252  4  63
 3  2  49  4  49 7
 3  23  7 2  22  32  7
HCF  22  3  7

12. Real Numbers
1. Prime Factors
Every natural number can be written as a product of its prime factors.
e.g. 324  4  81
 22  34
2. Highest Common Factor (HCF)
1) Write both numbers in terms of its prime factors
2) Take out the common factors
e.g. 1176 and 252
1176  6 196 252  4  63
 3  2  49  4  49 7
 3  23  7 2  22  32  7
HCF  22  3  7
 84

13. Real Numbers
1. Prime Factors
Every natural number can be written as a product of its prime factors.
e.g. 324  4  81
 22  34
2. Highest Common Factor (HCF)
1) Write both numbers in terms of its prime factors
2) Take out the common factors
e.g. 1176 and 252
1176  6 196 252  4  63
 3  2  49  4  49 7
 3  23  7 2  22  32  7
HCF  22  3  7 When factorising, remove
 84 the lowest power

14. 3. Lowest Common Multiple (LCM)

15. 3. Lowest Common Multiple (LCM)
1) Write both numbers in terms of its prime factors

16. 3. Lowest Common Multiple (LCM)
1) Write both numbers in terms of its prime factors
2) Write down all factors without repeating

17. 3. Lowest Common Multiple (LCM)
1) Write both numbers in terms of its prime factors
2) Write down all factors without repeating
e.g. 48 and 15

18. 3. Lowest Common Multiple (LCM)
1) Write both numbers in terms of its prime factors
2) Write down all factors without repeating
e.g. 48 and 15
48  16  3
 24  3

19. 3. Lowest Common Multiple (LCM)
1) Write both numbers in terms of its prime factors
2) Write down all factors without repeating
e.g. 48 and 15
48  16  3 15  3  5
 24  3

20. 3. Lowest Common Multiple (LCM)
1) Write both numbers in terms of its prime factors
2) Write down all factors without repeating
e.g. 48 and 15
48  16  3 15  3  5
 24  3
LCM  24  3  5

21. 3. Lowest Common Multiple (LCM)
1) Write both numbers in terms of its prime factors
2) Write down all factors without repeating
e.g. 48 and 15
48  16  3 15  3  5
 24  3
LCM  24  3  5
 240

22. 3. Lowest Common Multiple (LCM)
1) Write both numbers in terms of its prime factors
2) Write down all factors without repeating
e.g. 48 and 15
48  16  3 15  3  5
 24  3
When creating a LCM,
LCM  2  3  5
4
use the highest power
 240

23. 3. Lowest Common Multiple (LCM)
1) Write both numbers in terms of its prime factors
2) Write down all factors without repeating
e.g. 48 and 15
48  16  3 15  3  5
 24  3
When creating a LCM,
LCM  2  3  5
4
use the highest power
 240
4. Divisibility Tests

24. 3. Lowest Common Multiple (LCM)
1) Write both numbers in terms of its prime factors
2) Write down all factors without repeating
e.g. 48 and 15
48  16  3 15  3  5
 24  3
When creating a LCM,
LCM  2  3  5
4
use the highest power
 240
4. Divisibility Tests
2: even number

25. 3. Lowest Common Multiple (LCM)
1) Write both numbers in terms of its prime factors
2) Write down all factors without repeating
e.g. 48 and 15
48  16  3 15  3  5
 24  3
When creating a LCM,
LCM  2  3  5
4
use the highest power
 240
4. Divisibility Tests
2: even number
3: digits add to a multiple of 3

26. 3. Lowest Common Multiple (LCM)
1) Write both numbers in terms of its prime factors
2) Write down all factors without repeating
e.g. 48 and 15
48  16  3 15  3  5
 24  3
When creating a LCM,
LCM  2  3  5
4
use the highest power
 240
4. Divisibility Tests
2: even number
3: digits add to a multiple of 3
4: last two digits are divisible by 4

27. 3. Lowest Common Multiple (LCM)
1) Write both numbers in terms of its prime factors
2) Write down all factors without repeating
e.g. 48 and 15
48  16  3 15  3  5
 24  3
When creating a LCM,
LCM  2  3  5
4
use the highest power
 240
4. Divisibility Tests
2: even number
3: digits add to a multiple of 3
4: last two digits are divisible by 4
5: ends in a 5 or 0

28. 3. Lowest Common Multiple (LCM)
1) Write both numbers in terms of its prime factors
2) Write down all factors without repeating
e.g. 48 and 15
48  16  3 15  3  5
 24  3
When creating a LCM,
LCM  2  3  5
4
use the highest power
 240
4. Divisibility Tests
2: even number
3: digits add to a multiple of 3
4: last two digits are divisible by 4
5: ends in a 5 or 0
6: divisible by 2 and 3

29. 3. Lowest Common Multiple (LCM)
1) Write both numbers in terms of its prime factors
2) Write down all factors without repeating
e.g. 48 and 15
48  16  3 15  3  5
 24  3
When creating a LCM,
LCM  2  3  5
4
use the highest power
 240
4. Divisibility Tests
2: even number
3: digits add to a multiple of 3
4: last two digits are divisible by 4
5: ends in a 5 or 0
6: divisible by 2 and 3
7: double the last digit and subtract from
the other digits, answer is divisible by 7

30. 3. Lowest Common Multiple (LCM)
1) Write both numbers in terms of its prime factors
2) Write down all factors without repeating
e.g. 48 and 15
48  16  3 15  3  5
 24  3
When creating a LCM,
LCM  2  3  5
4
use the highest power
 240
4. Divisibility Tests
2: even number
3: digits add to a multiple of 3
4: last two digits are divisible by 4
5: ends in a 5 or 0
6: divisible by 2 and 3
7: double the last digit and subtract from
the other digits, answer is divisible by 7

31. 3. Lowest Common Multiple (LCM)
1) Write both numbers in terms of its prime factors
2) Write down all factors without repeating
e.g. 48 and 15
48  16  3 15  3  5
 24  3
When creating a LCM,
LCM  2  3  5
4
use the highest power
 240
4. Divisibility Tests
2: even number 8: last three digits are divisible by 8
3: digits add to a multiple of 3
4: last two digits are divisible by 4
5: ends in a 5 or 0
6: divisible by 2 and 3
7: double the last digit and subtract from
the other digits, answer is divisible by 7

32. 3. Lowest Common Multiple (LCM)
1) Write both numbers in terms of its prime factors
2) Write down all factors without repeating
e.g. 48 and 15
48  16  3 15  3  5
 24  3
When creating a LCM,
LCM  2  3  5
4
use the highest power
 240
4. Divisibility Tests
2: even number 8: last three digits are divisible by 8
3: digits add to a multiple of 3 9: sum of the digits is divisible by 9
4: last two digits are divisible by 4
5: ends in a 5 or 0
6: divisible by 2 and 3
7: double the last digit and subtract from
the other digits, answer is divisible by 7

33. 3. Lowest Common Multiple (LCM)
1) Write both numbers in terms of its prime factors
2) Write down all factors without repeating
e.g. 48 and 15
48  16  3 15  3  5
 24  3
When creating a LCM,
LCM  2  3  5
4
use the highest power
 240
4. Divisibility Tests
2: even number 8: last three digits are divisible by 8
3: digits add to a multiple of 3 9: sum of the digits is divisible by 9
4: last two digits are divisible by 4 10: ends in a 0
5: ends in a 5 or 0
6: divisible by 2 and 3
7: double the last digit and subtract from
the other digits, answer is divisible by 7

34. 3. Lowest Common Multiple (LCM)
1) Write both numbers in terms of its prime factors
2) Write down all factors without repeating
e.g. 48 and 15
48  16  3 15  3  5
 24  3
When creating a LCM,
LCM  2  3  5
4
use the highest power
 240
4. Divisibility Tests
2: even number 8: last three digits are divisible by 8
3: digits add to a multiple of 3 9: sum of the digits is divisible by 9
4: last two digits are divisible by 4 10: ends in a 0
5: ends in a 5 or 0 11: sum of even positioned digits =
6: divisible by 2 and 3 sum of odd positioned digits, or
differ by a multiple of 11.
7: double the last digit and subtract from
the other digits, answer is divisible by 7

35. Fractions & Decimals

36. Fractions & Decimals
Converting Recurring Decimals into Fractions

e.g.(i ) 0.6

37. Fractions & Decimals
Converting Recurring Decimals into Fractions

e.g.(i ) 0.6  0.666666

38. Fractions & Decimals
Converting Recurring Decimals into Fractions

e.g.(i ) 0.6  0.666666

let x  0.6
x  0.666666

39. Fractions & Decimals
Converting Recurring Decimals into Fractions

e.g.(i ) 0.6  0.666666

let x  0.6
x  0.666666
10 x  6.666666

40. Fractions & Decimals
Converting Recurring Decimals into Fractions

e.g.(i ) 0.6  0.666666

let x  0.6
x  0.666666
10 x  6.666666
9x  6

41. Fractions & Decimals
Converting Recurring Decimals into Fractions

e.g.(i ) 0.6  0.666666

let x  0.6
x  0.666666
10 x  6.666666
9x  6
x
6 2
 0.6
9 3

42. Fractions & Decimals
Converting Recurring Decimals into Fractions

e.g.(i ) 0.6  0.666666 
(ii ) 0.81  0.818181

let x  0.6
x  0.666666
10 x  6.666666
9x  6
x
6 2
 0.6
9 3

43. Fractions & Decimals
Converting Recurring Decimals into Fractions

e.g.(i ) 0.6  0.666666 
(ii ) 0.81  0.818181

let x  0.6 
let x  0.81
x  0.666666 x  0.818181
10 x  6.666666
9x  6
x
6 2
 0.6
9 3

44. Fractions & Decimals
Converting Recurring Decimals into Fractions

e.g.(i ) 0.6  0.666666 
(ii ) 0.81  0.818181

let x  0.6 
let x  0.81
x  0.666666 x  0.818181
10 x  6.666666 100 x  81.818181
9x  6
x
6 2
 0.6
9 3

45. Fractions & Decimals
Converting Recurring Decimals into Fractions

e.g.(i ) 0.6  0.666666 
(ii ) 0.81  0.818181

let x  0.6 
let x  0.81
x  0.666666 x  0.818181
10 x  6.666666 100 x  81.818181
9x  6 99 x  81
x
6 2
 0.6
9 3

46. Fractions & Decimals
Converting Recurring Decimals into Fractions

e.g.(i ) 0.6  0.666666 
(ii ) 0.81  0.818181

let x  0.6 
let x  0.81
x  0.666666 x  0.818181
10 x  6.666666 100 x  81.818181
9x  6 99 x  81
x
6 2
 0.6 x
81  9
 0.81 
9 3 99 11

47. Fractions & Decimals
Converting Recurring Decimals into Fractions

e.g.(i ) 0.6  0.666666 
(ii ) 0.81  0.818181

let x  0.6 
let x  0.81
x  0.666666 x  0.818181
10 x  6.666666 100 x  81.818181
9x  6 99 x  81
x
6 2
 0.6 x
81  9
 0.81 
9 3 99 11

(iii ) 0.327  0.3272727

48. Fractions & Decimals
Converting Recurring Decimals into Fractions

e.g.(i ) 0.6  0.666666 
(ii ) 0.81  0.818181

let x  0.6 
let x  0.81
x  0.666666 x  0.818181
10 x  6.666666 100 x  81.818181
9x  6 99 x  81
x
6 2
 0.6 x
81  9
 0.81 
9 3 99 11

(iii ) 0.327  0.3272727

let x  0.327
x  0.3272727

49. Fractions & Decimals
Converting Recurring Decimals into Fractions

e.g.(i ) 0.6  0.666666 
(ii ) 0.81  0.818181

let x  0.6 
let x  0.81
x  0.666666 x  0.818181
10 x  6.666666 100 x  81.818181
9x  6 99 x  81
x
6 2
 0.6 x
81  9
 0.81 
9 3 99 11

(iii ) 0.327  0.3272727

let x  0.327
x  0.3272727
100 x  32.7272727

50. Fractions & Decimals
Converting Recurring Decimals into Fractions

e.g.(i ) 0.6  0.666666 
(ii ) 0.81  0.818181

let x  0.6 
let x  0.81
x  0.666666 x  0.818181
10 x  6.666666 100 x  81.818181
9x  6 99 x  81
x
6 2
 0.6 x
81  9
 0.81 
9 3 99 11

(iii ) 0.327  0.3272727

let x  0.327
x  0.3272727
100 x  32.7272727
99 x  32.4

51. Fractions & Decimals
Converting Recurring Decimals into Fractions

e.g.(i ) 0.6  0.666666 
(ii ) 0.81  0.818181

let x  0.6 
let x  0.81
x  0.666666 x  0.818181
10 x  6.666666 100 x  81.818181
9x  6 99 x  81
x
6 2
 0.6 x
81  9
 0.81 
9 3 99 11

(iii ) 0.327  0.3272727

let x  0.327
x  0.3272727
100 x  32.7272727
99 x  32.4
x
32.4 324
    18
 0.327
99 990 55

52. Alternatively:

e.g.(i ) 0.6

53. Alternatively:
 6 6 is recurring
e.g.(i ) 0.6

54. Alternatively:
 6 6 is recurring
e.g.(i ) 0.6
9 1 number recurring,
2 use ‘9’

3

55. Alternatively:
 6 6 is recurring
e.g.(i ) 0.6
9 1 number recurring,
2 use ‘9’

3

(ii ) 0.81

56. Alternatively:
 6 6 is recurring
e.g.(i ) 0.6
9 1 number recurring,
2 use ‘9’

3
   81 81 is recurring
(ii ) 0.81

57. Alternatively:
 6 6 is recurring
e.g.(i ) 0.6
9 1 number recurring,
2 use ‘9’

3
   81 81 is recurring
(ii ) 0.81
99 2 numbers recurring,
9 use ‘99’

11

58. Alternatively:
 6 6 is recurring
e.g.(i ) 0.6  
(iii ) 0.7134
9 1 number recurring,
2 use ‘9’

3
   81 81 is recurring
(ii ) 0.81
99 2 numbers recurring,
9 use ‘99’

11

59. Alternatively:
 6 6 is recurring
e.g.(i ) 0.6   7134
(iii ) 0.7134 
9 1 number recurring, 9999
2 use ‘9’
 
2378
3 3333
   81 81 is recurring
(ii ) 0.81
99 2 numbers recurring,
9 use ‘99’

11

60. Alternatively:
 6 6 is recurring
e.g.(i ) 0.6   7134
(iii ) 0.7134 
9 1 number recurring, 9999
2 use ‘9’
 
2378
3 3333
   81 81 is recurring
(ii ) 0.81
99 2 numbers recurring,
9 use ‘99’

11

(iv) 0.327

61. Alternatively:
 6 6 is recurring
e.g.(i ) 0.6   7134
(iii ) 0.7134 
9 1 number recurring, 9999
2 use ‘9’
 
2378
3 3333
   81 81 is recurring
(ii ) 0.81
99 2 numbers recurring,
9 use ‘99’

11
   324 327 – 3 ( subtract number not recurring)
(iv) 0.327

62. Alternatively:
 6 6 is recurring
e.g.(i ) 0.6   7134
(iii ) 0.7134 
9 1 number recurring, 9999
2 use ‘9’
 
2378
3 3333
   81 81 is recurring
(ii ) 0.81
99 2 numbers recurring,
9 use ‘99’

11
   324 327 – 3 ( subtract number not recurring)
(iv) 0.327
990 2 numbers recurring, 1 not
18 use ‘990’

55

63. Alternatively:
 6 6 is recurring
e.g.(i ) 0.6   7134
(iii ) 0.7134 
9 1 number recurring, 9999
2 use ‘9’
 
2378
3 3333
   81 81 is recurring
(ii ) 0.81
99 2 numbers recurring,
9 use ‘99’

11
   324 327 – 3 ( subtract number not recurring)
(iv) 0.327
990 2 numbers recurring, 1 not
18 use ‘990’

55

(v) 0.1096

64. Alternatively:
 6 6 is recurring
e.g.(i ) 0.6   7134
(iii ) 0.7134 
9 1 number recurring, 9999
2 use ‘9’
 
2378
3 3333
   81 81 is recurring
(ii ) 0.81
99 2 numbers recurring,
9 use ‘99’

11
   324 327 – 3 ( subtract number not recurring)
(iv) 0.327
990 2 numbers recurring, 1 not
18 use ‘990’

55
  1086 1096 – 10
(v) 0.1096 

65. Alternatively:
 6 6 is recurring
e.g.(i ) 0.6   7134
(iii ) 0.7134 
9 1 number recurring, 9999
2 use ‘9’
 
2378
3 3333
   81 81 is recurring
(ii ) 0.81
99 2 numbers recurring,
9 use ‘99’

11
   324 327 – 3 ( subtract number not recurring)
(iv) 0.327
990 2 numbers recurring, 1 not
18 use ‘990’

55
  1086 1096 – 10
(v) 0.1096 
9900 2 numbers recurring, 2 not
181 use ‘9900’

1650

66. Alternatively:
 6 6 is recurring
e.g.(i ) 0.6   7134
(iii ) 0.7134 
9 1 number recurring, 9999
2 use ‘9’
 
2378
3 3333
   81 81 is recurring
(ii ) 0.81
99 2 numbers recurring,
9 use ‘99’

11
   324 327 – 3 ( subtract number not recurring)
(iv) 0.327
990 2 numbers recurring, 1 not
18 Exercise 2A;
use ‘990’
 2adgj, 3bd, 4ac,
55
5acegi, 6, 7cdg,
  1086 1096 – 10
(v) 0.1096  8bdfhj, 9, 10bd,
9900 2 numbers recurring, 2 not 11ac, 12, 13*,
181 use ‘9900’
 14*
1650