Download Whole Number Arithmetic 98 DS

EXERCISE 1: (Place Value)
ORAL EXAMPLES: State these numbers in words.
1. 100
2. 1000
3.
5. 1 000 000
6. 10 000 000
7.
9. 900
10. 4000
11.
13. 8 000 000
14. 50 000 000
15.
WORKED EXAMPLE: Write 604 050 in an expanded form.
10 000
100 000 000
70 000
300 000 000
4.
8.
12.
16.
100 000
1 000 000 000
200 000
6 000 000 000
604 050 = 6 ´100 000 + 4 ´1000 + 5´10.
WRITTEN EXAMPLES: Write these numbers in an expanded form.
1. 825
2. 904
3. 370
4. 6159
5. 8430
6. 2097
7. 5608
8. 41 586
9. 70 243
10. 964 710
11. 309 060
12. 5 020 800
Write down the value of each 3 and each 7 in these numbers.
13. 19 375
14. 57 193
15. 532 764
16. 798 231
17. 9 213 867
18. 7 306 842
19. 3 074 586
20. 5 947 283
21. 693 750
22. 759 631
23. 5 436 827
24. 7 021 398
25. 19 384 572
26. 302 875 469
27. 573 264 891
28. 7 139 528 640
*29. There are 1000 mm in a metre and 1000 m in a kilometre.
(i) How many millimetres in a kilometre?
(ii) How many millimetres in 1000 km? (the distance from Auckland to Wellington and back).
*30. Write down the number (i) one billion (ii) one trillion
EXERCISE 2: (Order of Operations)
Note:
WORKED EXAMPLES:
1. Do the brackets first
a. (i) (20 +10) ´ 5 = 30 ´ 5 =150
(ii) 20 ¸ (10 - 5) = 20 ¸ 5 = 4
2. Do ´, ¸ before +, b. (i) 20 +10 ´ 5 = 20 + 50 = 70
(ii) 20 -10 ¸ 5 = 20 - 2 =18
3. Work from left to right
c. (i) 20 -10 + 5 =10 + 5 =15
(ii) 20 ¸10 ´ 5 = 2 ´ 5 =10
ORAL EXAMPLES: Find the value of
1. 10 + 5+ 2
2. 10 + 5- 2
3. 10 - 5+ 2
4. 10 - 5- 2
5. 10 ´ 5´ 2
6. 10 ´ 5¸ 2
7. 10 ¸ 5´ 2
8. 10 ¸ 5¸ 2
9. 10 ´ (5+ 2)
10. 10 ´ 5+ 2
11. (10 + 5) ´ 2
12. 10 + 5´ 2
13. 10 ´ (5- 2)
14. 10 ´ 5- 2
15. (10 - 5) ´ 2
16. 10 - 5´ 2
WRITTEN EXAMPLES: Find the value of
1. 8 + 4 + 2
2. 8 + 4 - 2
3. 8 - 4 + 2
4. 8 - (4 + 2)
5. 8 - 4 - 2
6. 8 - (4 - 2)
7. 8 ´ 4 ´ 2
8. 8 ´ 4 ¸ 2
9. 8 ¸ 4 ´ 2
10. 8 ¸ (4 ´ 2)
11. 8 ¸ 4 ¸ 2
12. 8 ¸ (4 ¸ 2)
13. 8 ´ (4 + 2)
14. 8 ´ 4 + 2
15. (8 + 4) ´ 2
16. 8 + 4 ´ 2
17. 8 ´ (4 - 2)
18. 8 ´ 4 - 2
19. (8 - 4) ´ 2
20. 8 - 4 ´ 2
21. 8 ¸ (4 - 2)
22. 8 ¸ 4 - 2
23. (8 - 4) ¸ 2
24. 8 - 4 ¸ 2
25. 12 + 6 + 2
26. 12 + 6 - 2
27. 12 - 6 + 2
28. 12 - (6 + 2)
29. 12 - 6 - 2
30. 12 - (6 - 2)
31. 12 ´ 6 ´ 2
32. 12 ´ 6 ¸ 2
33. 12 ¸ 6 ´ 2
34. 12 ¸ (6 ´ 2)
35. 12 ¸ 6 ¸ 2
36. 12 ¸ (6 ¸ 2)
37. 12 ´ (6 + 2)
38. 12 ´ 6 + 2
39. (12 + 6) ´ 2
40. 12 + 6 ´ 2
41. 12 ´ (6 - 2)
42. 12 ´ 6 - 2
43. (12 - 6) ´ 2
44. 12 - 6 ´ 2
45. 12 ¸ (6 - 2)
46. 12 ¸ 6 - 2
47. (12 - 6) ¸ 2
48. 12 - 6 ¸ 2
* Use 2 different operations (choosing from +,-,´,¸) to show how three 4's (i.e. the numbers 4, 4 and 4)
can be combined to give the following answers. Brackets can be used- some can be done in more than
one way.
49. 4
50. 20
51. 32
52. 12
53. 5
54. 2
55. 3
56. 0
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Page 1 of 19
EXERCISE 3: (Powers)
Note: 5´ 5´ 5 = 53(where 3 is called the power and 5 is called the base)
whereas 35 (in which 5 is the power and 3 is the base) = 3´ 3´ 3´ 3´ 3
ORAL EXAMPLES:
State these in shorthand.
5´ 5´ 5´ 5´ 5´ 5
6´6´6´6´6
1.
2.
8´8´8
3´ 3´ 3´ 3´ 3´ 3´ 3´ 3
3.
4.
State these in long hand.
5.
29
92
6.
74
47
7.
8.
WORKED EXAMPLES:
Evaluate a. 93
b. 8100
3
a. 9 = 9 ´ 9 ´ 9 = 81´ 9 = 729
b. 902 = 90 ´ 90 = 9 ´10 ´ 9 ´10 = 8100
Hence 8100 = 90
WRITTEN EXAMPLES: Evaluate (i.e. find the value of )
1. 22
2. 23
3. 2 4
4. 2 5
5. 26
6. 2 7
7. 28
8. 29
9. 62
10. 72
11. 82
12. 92
13. 32
14. 33
15. 34
16. 35
17. 42
18. 43
19. 52
20. 53
21. 102
22. 103
23. 10 4
24. 105
25. 106
26. 1002
27. 1003
28. 10002
29. 202
30. 203
31. 502
32. 503
33. 302
34. 402
35. 3002
36. 4002
37. 12
38. 13
39. 14
40. 15
41. 21
42. 31
43. 41
44. 51
45. 22 ´ 52
46. 23 ´ 53
47. 23 ´ 32
48. 22 ´ 33
49. 5 ´ 22
50. 2 ´ 52
51. 4 ´ 32
52. 3 ´ 42
53. 9
54.
55. 36
56.
81
64
57.
58.
59.
60.
4
16
25
49
61.
62.
63.
64.
10 000
1 000 000
1
100
400
2
69. (7 + 3)
65.
73.
77.
(16 + 9)
52
900
70. 7 + 32
66.
2
74.
78.
16 + 9
72
1600
2
71. (6 - 4)
67.
75.
79.
(100 - 64)
( )
2500
72. 6 - 42
68.
2
76.
2
4
*Find the smallest value of n if n is a whole number and
81. 2n >100
82. 3n >100
83. 2n >1000
2
3
85. n > 100
86. n > 100
87. n 2 > 1000
80.
100 - 64
( 6)
2
84. 3n >1000
88. n 3 > 1000
Copy down and complete these statements using patterns to help you find the value of
20, 50 and 100.
89. 23 = 2 ´ 2 ´ 2=
90. 53 = 5´ 5´ 5 =
91. 103 =10 ´10 ´10 =_____
22 = 2 ´ 2=
52 = 5´ 5= __
102 =10 ´10 = ______
21 =________
51 =
101 =_______
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20 = ________
EXERCISE 4:
50 =_______
100 = ________
(Working with Powers)
Note: 1. The division symbol can be replaced by a “quotient bar” e.g. 6 ¸ 2 =
6
=3
2
5 + 3 (5 + 3) 8
=
= = 8¸4 = 2
4
4
4
4
3
2. 2 ´ 2 = (2 ´ 2 ´ 2 ´ 2) ´ (2 ´ 2 ´ 2) = 2 ´ 2 ´ 2 ´ 2 ´ 2 ´ 2 ´ 2 = 27 = 24+3
The quotient bar acts as a bracket e.g.
(
)
2 2 ´ 2 ´ 2 ´ 2 ´ 2/ ´ 2/ ´ 2/
=
= 2 ´ 2 ´ 2 ´ 2 = 2 4 = 2 7-3
3
/
/
/
2
´
2
´
2
2
Hence, as long as the powers have the same base (in this case 2) then to multiply we can just add the powers
(4 and 3) and to divide we can just subtract the powers (7 and 3).
ORAL EXAMPLES: State these products and quotients as powers of 2, 3 or 5.
1. 23 ´ 23
2. 26 ¸ 23
3. 34 ´ 32
4. 36 ¸ 32
5. 24 ´ 24
6. 28 ¸ 24
7. 310 ´ 32
8. 312 ¸ 32
9. 56 ´ 52
10. 58 ¸ 52
11. 59 ´ 53
12. 512 ¸ 53
2 7 ¸ 23 =
(
7
)
WORKED EXAMPLES: Write these products and quotients as powers of 2, 3 or 5.
5 9 ´ 53 51 2
a. 210 x 25 = 210 + 5 = 215
b. 315 ¸ 35 = 315 – 5 = 310
c.
= 6 = 56
6
5
5
WRITTEN EXAMPLES: Write these products and quotients as powers of 2, 3 or 5.
1. 28 ´ 24
2. 212 ¸ 24
3. 36 ´ 36
4. 312 ¸ 36
5. 58 ´ 52
6. 510 ¸ 52
7. 25 ´ 25
8. 210 ¸ 25
9. 36 ´ 33
10. 39 ¸ 33
11. 56 ´ 5
12. 57 ¸ 5
6
4
6
4
12
4
2 ´2
3 ´3
2 ´2
51 2 ´ 5 6
13.
14
15.
16.
28
32
28
59
Table of Powers
21 = 2
22 = 4
23 = 8
24 =16
25 = 32
26 = 64
27 =128
28 = 256
29 = 512
210 =1024
211 = 2048
212 = 4096
213 = 8192
214 =16 384
215 = 32 768
216 = 65 536
217 =131 072
218 = 262 144
219 = 524 288
220 =1048 576
31 = 3
32 = 9
33 = 27
34 = 81
35 = 243
36 = 729
37 = 2187
38 = 6561
39 =19 683
310 = 59 049
311 =177 147
312 = 531 441
51 = 5
52 = 25
53 =125
54 = 625
55 = 3125
56 =15 625
57 = 78 125
58 = 390 625
59 =1953125
510 = 9 765 625
WORKED EXAMPLES:
Use the table of powers to find these products and quotients.
d. 6561´ 81= 38 ´ 34 = 312 = 531 441
e. 6561¸ 81= 38 ¸ 34 = 34 = 81
WRITTEN EXAMPLES (continued): Use the table of powers to find these products and quotients.
17. 256 ´16
18. 512 ´ 32
19. 1024 ´ 64
20. 2048 ´128
21. 8192 ¸ 512
22. 32 768 ¸1024
23. 131 072 ¸ 2048
24. 524 288 ¸ 4096
25. 729 ´ 27
26. 2187 ´ 81
27. 531 441¸ 6561
28. 59 049 ¸ 243
29. 625 ´ 25
30. 3125 ´125
31. 78 125 ¸ 625
32. 9 765 625 ¸ 25
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*33.
16 384 ´ 4 096
262 144
*34.
65 536 ´ 8192
1 048 576
*35.
19 683 ´ 2187
177 147
*36.
1 953 125 ´15 625
390 625
EXERCISE 5: (Factors, Prime Numbers, Multiples)
Note: 2 is a factor of 6 (because 2 divides evenly into 6) and so 6 is a multiple of 2.
4 is not a factor of 6 and so 6 is not a multiple of 4.
WORKED EXAMPLES:
Roster (i.e. list) the elements of
a. { factors of 8 }
8 =1´ 8 = 2 ´ 4 and so { factors of 8 } = { 1, 2, 4, 8 }
b. { factors of 100 }
100 =1´100 = 2 ´ 50 = 4 ´ 25 = 5´ 20 =10 ´10 and so
{ factors of 100 } = { 1, 2, 4, 5, 10, 20, 25, 50, 100 }
Note: 1. It does not matter in which order the elements are rostered.
e.g. { 1, 2, 4, 8 } = { 1, 8, 2, 4 }.
2. The elements of a set are not to be repeated.
ORAL EXAMPLES: State the elements of these sets.
1. { factors of 15 }
2. { factors of 32 }
3. { factors of 27 }
4. { factors of 28 }
WRITTEN EXAMPLES: Roster the elements of these sets.
1. { factors of 4 }
2. { factors of 9 }
3. { factors of 16 }
4. { factors of 25 }
5. { factors of 36 }
6. { factors of 1 }
7. { factors of 6 }
8. { factors of 12 }
9. { factors of 18 }
10. { factors of 24 }
11. { factors of 10 }
12. { factors of 20 }
13. { factors of 30 }
14. { factors of 40 }
15. { factors of 2 }
16. { factors of 3 }
17. { factors of 5 }
18. { factors of 7 }
19. { factors of 11 }
20. { factors of 13 }
Note:
1.
2.
3.
Prime numbers have exactly 2 factors ( namely 1 and itself).
e.g. The numbers in questions 15 to 20 are prime numbers.
If a factor is a prime number then it is called a prime factor.
e.g. { factors of 100 } = {1, 2, 4, 5, 10, 20, 25, 50, 100 }
{ prime factors of 100 } = { 2, 5 }
The number 1 is not a prime number and so it is not a prime factor of any number.
WRITTEN EXAMPLES (continued): Roster the elements of these sets.
21. { prime numbers between 0 and 10 }
22. { prime numbers between 10 and 20 }
23. { prime numbers between 20 and 30 }
24. { prime numbers between 30 and 40 }
25. { prime factors of 6 }
26. { prime factors of 10 }
27. { prime factors of 14 }
28. { prime factors of 15 }
29. { prime factors of 21 }
30. { prime factors of 35 }
31. { prime factors of 30 }
32. { prime factors of 42 }
Note: Sets of factors are finite sets whereas sets of multiples are infinite sets and so all the elements cannot
be rostered. In such cases we can roster sufficient elements for the pattern to be clear and then put 3
dots for the words "and so on.".
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Page 4 of 19
WORKED EXAMPLE: c. Roster the elements of { multiples of 5 }.{ multiples of 5 } = { 5, 10, 15... }
WRITTEN EXAMPLES (continued): Roster the elements of these sets.
33. { multiples of 3 }
34. { multiples of 6 }
35. { multiples of 2 }
36. { multiples of 4 }
Copy down and complete these statements.
37. The only even number which is prime is the number___.
38. All numbers have at least 2 factors except for the number
.
39. All prime numbers have exactly ___ factors.
40. The square numbers (i.e. 1, 4, 9, 16, 25, 36, ... ) have an
number of factors, whereas all
other numbers have an
__ number of factors.
*Roster the elements of these sets.
41. { factors of 60 }
42. { factors of 360 }
43. { prime numbers between 40 and 50 }
44. { prime numbers between 50 and 60 }
EXERCISE 6: (Prime Factors)
Note:
The index notation is useful in this Exercise.
e.g. 2 ´ 2 ´ 2 = 23, 3´ 3 = 32 and so 2 ´ 2 ´ 2 ´ 3´ 3 = 23 ´ 32
WORKED EXAMPLES: Write these numbers as the product of their prime factors.
a. 42 = 6 ´ 7 = 2 ´ 3´ 7 or 2 ´ 21= 2 ´ 3´ 7
b. 28 = 4 ´ 7 = 22 ´ 7
c. 64 = 8 ´ 8 = 2 ´ 2 ´ 2 ´ 2 ´ 2 ´ 2 = 26
d. 72 = 8 ´ 9 = 2 ´ 2 ´ 2 ´ 3´ 3 = 23 ´ 32
WRITTEN EXAMPLES: Write these numbers as the product of their prime factors.
1. 6
2. 10
3. 14
4.
5. 21
6. 35
7. 30
8.
9. 4
10. 8
11. 16
12.
13. 9
14. 27
15. 25
16.
17. 12
18. 18
19. 20
20.
21. 45
22. 75
23. 36
24.
25. 24
26. 54
27. 40
28.
29. 48
30. 80
31. 90
32.
*33.
*34.
*35.
*36.
15
70
32
49
50
60
56
84
Find the smallest number which is the product of 4 different prime factors.
Find the next smallest number which is the product of 4 different prime factors.
Find the smallest number which is the product of 4 prime factors (not necessarily different).
Find the next smallest number which is the product of 4 prime factors (not necessarily different).
WORKED EXAMPLES:
Find the highest common factor (H.C.F.) and the lowest common
multiple (L.C.M.) of 8 and 12.
e. { factors of 8 } = { 1, 2, 4, 8 }
{ factors of 12 } = { 1, 2, 3, 4, 6, 12 }
H.C.F. = 4
f. { multiples of 8 } = { 8, 16, 24, 32, 40, 48, ... }
{ multiples of 12 } = { 12, 24, 36, 48, 60, ... }
L.C.M. = 24
WRITTEN EXAMPLES (continued): State the H.C.F. and L.C.M. of
37. 2 and 6
38. 2 and 10
39. 3 and 6
41. 4 and 6
42. 6 and 8
43. 6 and 9
45. 2, 4 and 8
46. 4, 6 and 8
47. 3, 6 and 12
49. 10 and 15
50. 20 and 25
51. 20 and 30
40.
44.
48.
52.
3 and 12
9 and 12
6, 9 and 12
40 and 60
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EXERCISE 7:
Note:
1.
2.
3.
4.
( Set Notation )
If x is an element (or member) of a set A then we write x A.
If x is not an element of a set A then we write x A.
e.g. if A = { factors of 10 } = { 1, 2, 5, 10 } then 2 A, whereas 4 A.
If a set A has the same elements as a set B then we say A equals B and write A = B.
e.g. if A = { prime factors of 10 } = { 2, 5 }
and B = { prime factors of 20 } = { 2, 5 } , then A = B
If all the elements of A are also elements of B then we say
B 10
that A is a subset of B and write A  B.
1 A2
e.g. If A = {prime factors of 10} = {2, 5}
5
and B = {factors of 10} = {1, 2, 5, 10} then A  B.
connects an element to a set, whereas
connects a set to a set.
e.g. if A = { factors of 4 } = { 1, 2, 4 } then
2 A whereas { 2 } A ( 3 A and { 3 } A are also correct statements.).
2 A and { 2 } A are incorrect statements.
ORAL EXAMPLES: Say whether these statements are True or False.
1. 5 is a factor of 10.
2. 5 is a multiple of 10.
3. 10 is a factor of 5.
4. 10 is a multiple of 5.
5. 7 is a factor of 7.
6. 7 is a multiple of 7.
7. 7 is a prime number.
8. 9 is a prime number.
9. 3 is a prime factor of 12.
WRITTEN EXAMPLES: Choose from , , or to complete these statements.
1. 2
{ 1, 2, 3 }
2. { 2 }
{ 1, 2, 3 }
3. 7
{ 4, 5, 6 }
4. { 7 }
{ 4, 5, 6 }
5. 3
{ 2, 3, 4 }
6. 8
{ 5, 6, 7 }
7. { 3 }
{ 2, 3, 4 }
8. { 8 }
{ 5, 6, 7 }
9. 5
{ factors of 5 }
10. 5
{ multiples of 5 }
11. 4
{ factors of 8 }
12. 4
{ multiples of 8 }
13. 6
{ factors of 3 }
14. 6
{ multiples of 3 }
15. { 5 }
{ factors of 5 }
16. { 5 }
{ multiples of 5 }
17. { 4 }
{ factors of 8 }
18. { 4 }
{ multiples of 8 }
19. { 6 }
{ factors of 3 }
20. { 6 }
{ multiples of 3 }
21. 17
{ prime numbers }
22. { 17 }
{ prime numbers }
23. 27
{ prime numbers }
24. { 27 }
{ prime numbers }
25. 1
{ prime factors of 3 }
26. 2
{ prime factors of 6 }
27. { 3 }
{ prime factors of 9 }
28. { 4 }
{ prime factors of 12 }
29. If A = { factors of 2 }, B = { factors of 3 }, C = { factors of 4 }, D = { factors of 6 },
E = { factors of 8 } and F = { factors of 9 }.
a) Roster the elements of each set.
b) Write down which set is a subset of another set ( e.g. A C ).
* 30. If P = { prime factors of 18 }, Q = { prime factors of 24 }, R = { prime factors of 30 },
S = { prime factors of 35 }, T = { prime factors of 45 }, U = { prime factors of 60 } and
V = { prime factors of 70 }.
a) Roster the elements of each set. (Note: They all have either 2 or 3 elements.).
b) Write down which sets are equal.
c) Write down which set is a subset of another set, but not equal to it.
Note :
1.
N = { natural numbers } = { 1, 2, 3, 4, 5, ... }
W = { whole numbers } = { 0, 1, 2, 3, 4, ... }
1, 2,
3,... N W
2. In the above Exercise, the universal set, usually written as , was the set of natural numbers,
N={1,2,3,4,..} This means that the only numbers available when answering questions were
the elements of N.
Thus 6 =1´ 6 = 2 ´ 3 = 12 ´12 = etc.
0
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Page 6 of 19
1
{ factors of 6 } = { 1, 2, 3, 6 } and 2
1
{ factors of 6 }, because 2
N ( the universal set ).
EXERCISE 8: (Cancelling Factors)
WORKED EXAMPLES:
a.
b.
c.
17 ´ 19 17 ´ 19
=
= 17
19
19
17 ´19 17 ´19
=
=19
17
17
8 ´ 9 4/ ´ 2 ´ 3´ 3/
=
= 2 ´ 3 = 6 or
4/ ´ 3/
12
or
8/ ´ 9 2 ´ 9
=
=2´3=6
1/ 2/ 3
3
8 ´ 9/ 3 2 8/ ´ 3
=
=
= 2´3= 6
1/ 2/ 4
4/
=
2
27 ´ 32 9/ ´ 3 ´ 4 ´ 8/
=
= 3 ´ 4 = 12
72
9/ ´ 8/
35 ´ 54 7/ ´ 5/ ´ 6 ´ 9/ 6
e.
=
= =3
45 ´14 9/ ´ 5/ ´ 2 ´ 7/ 2
ORAL EXAMPLES: Simplify by cancelling factors.
7´5
7´5
11´13
11´13
1.
2.
3.
4.
5
7
13
11
6´9
6´9
7´8
7´8
5.
6.
7.
8.
9´2
9´3
2´7
4´7
8´6
8´3
6´9
2´9
9.
10.
11.
12.
3
6
2
6
WRITTEN EXAMPLES: Simplify by cancelling factors.
6´8
7 ´12
8 ´10
9 ´14
1.
2.
3.
4.
12
14
16
18
6´9
7 ´15
8 ´ 24
9 ´16
5.
6.
7.
8.
18
21
32
36
8 ´12
9 ´18
7 ´ 25
6 ´ 21
9.
10.
11.
12.
24
27
35
42
8 ´15
15 ´ 28
8 ´ 21
10 ´ 36
13.
14.
15.
16.
20
35
28
45
15 ´ 32
18 ´ 35
8 ´ 27
14 ´ 28
17.
18.
19.
20.
40
42
36
49
20 ´ 27
18 ´ 40
16 ´15
14 ´ 36
21.
22.
23.
24.
45
48
40
63
14 ´ 20
18 ´ 28
18 ´ 45
21´ 32
25.
26.
27.
28.
35
42
54
56
18 ´ 35
56 ´ 54
32 ´ 35
32 ´ 45
29.
30.
31.
32.
14 ´15
21´ 24
40 ´ 28
40 ´18
63 ´ 48
42 ´ 72
24 ´ 35
56 ´ 45
33.
34.
35.
36.
28 ´ 27
63 ´ 24
42 ´ 20
36 ´ 35
d.
* WORKED EXAMPLE:
f.
31 815 31 815 4545
=
=
= 505
63
7´9
9
* WRITTEN EXAMPLES (continued):
1365
1722
37. 15
38. 21
Find these quotients by cancelling factors.
2044
2048
39. 28
40. 32
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Page 7 of 19
41.
17 052
42
42.
13 815
45
43.
11 232
54
44.
6104
56
EXERCISE 9: ( Rounding Off )
Copy down the tables below, but round off as follows;
1. the lengths of N. Z. Rivers to the nearest 10 Km.
2. the heights of N. Z. Mountains to the nearest 100 m.
3. the areas of N. Z. Lakes to the nearest 1000 ha.
4. the areas of N. Z. Regions correct to 2 significant figures.
5. the population of N. Z. Regions correct to 3 significant figures.
River
Waikato
Clutha
Wanganui
Taieri
Rangitiki
Waitaki
Length
(in Km)
425
322
290
288
241
209
Mountain
Cook
Tasman
Ruapehu
Taranaki
Ngauruhoe
Tongariro
Region
Northland
Auckland
Waikato
Bay of Plenty
Gisborne
Hawkes' Bay
Taranaki
Manawatu - Wanganui
Wellington
Nelson
Tasman
Marlborough
West Coast
Canterbury
Otago
Southland
New Zealand
Height
(in m)
3764
3498
2797
2518
2291
1968
Area (in Km2 )
13 941
5 600
25 598
12 447
8 351
14 164
7 273
22 215
8 124
445
9 786
12 484
23 336
45 346
31 990
34 347
275 446
Lake
Taupo
Te Anau
Wakatipu
Wanaka
Manapouri
Hawea
Area
(in ha)
60 606
34 447
29 267
19 166
14 245
11 914
Population (1996)
137 052
1 068 645
350 125
224 365
45 787
142 789
106 589
228 770
414 048
40 279
37 973
38 397
32 512
468 040
185 083
97 100
3 618 302
6. Copy down and complete this table.
Year
1881
1901
1921
1951
1961
1971
1981
1991
N.Z.
Population
nearest
1000
Population rounded off
nearest
3 sig. fig.
10 000
to
2 sig. fig.
534 030
815 862
1 271 668
1 939 472
2 414 984
2 862 631
3 175 737
3 434 950
* 7. This table shows the population of Auckland's 4 cities rounded to the nearest 1000.
Copy down and complete the table. ( The first one has been done for you. )
DS/GRB/JMH: June 19, 2017
Page 8 of 19
City
North Shore
Waitakere
Auckland
Manukau
Population
172 000
156 000
346 000
254 000
Minimum Pop.
171 500
Maximum Pop.
172 499
EXERCISE 10: ( Approximate Calculations )
ORAL EXAMPLES:
1. Given that 9 ´ 6 = 54, state the value of
a. 90 ´ 6
b. 90 ´ 60
c. 900 ´ 60
2. Given that 8 ´ 5 = 40, state the value of
a. 80 ´ 5
b. 80 ´ 50
c. 800 ´ 50
3. Given that 40 ¸ 8 = 5, state the value of
400
4000
40 000
a.
b.
c.
8
80
80
4. Given that 80 ¸ 4 = 20, state the value of
800
8000
80 000
a.
b.
c.
4
40
40
WRITTEN EXAMPLES: Find these products and quotients.
1. 80 ´ 7
2. 80 ´ 70
3. 800 ´ 70
5. 40 ´ 5
6. 40 ´ 50
7. 400 ´ 50
3000
30 000
300
60
9. 6
10. 60
11.
600
6000
60 000
13.
14.
15.
3
30
30
d. 900 ´ 600
d. 800 ´ 500
d.
400 000
800
d.
800 000
400
4. 800 ´ 700
8. 400 ´ 500
300 000
600
12.
16.
600 000
300
WORKED EXAMPLES:
Round off each number correct to one significant figure and then calculate.
396 ~ 400
~ 400 x 50 = 20 000
a. 396 x 52 ~
b. 52 ~
50 = 8
2
2
c. (195) » (200) = 200 ´ 200 = 40 000
d. 41 325 » 40 000 = 200
WRITTEN EXAMPLES (continued):
Round off each number correct to one significant figure and then calculate.
17. 91´18
18. 82 ´ 29
19. 73´ 36
20. 64 ´ 47
21. 621´19
22. 685 ´ 32
23. 817 ´ 38
24. 893 ´ 51
513
123
196
586
25. 106
26. 54
27. 48
28. 192
786
604
597
4253
29. 19
30. 21
31. 28
32. 82
33. (18)2
34. (51)2
35. (302)2
36. (389)2
37.
104
38. 396
39. 915
40. 9810
78 ´ 21
64 ´ 29
91´ 38
39 ´ 62
41.
42.
43.
44.
37
93
59
81
82 ´ 27
63 ´ 37
87 ´ 23
92 ´ 41
45.
46.
47.
48.
57 ´ 22
18 ´ 32
26 ´14
23 ´ 61
49. (21)3
50. (29)3
51. (18) 4
52. (32) 4
* 53. The sun is 150 million kilometres from the earth. Light travels a distance of 300 000 kilometres
every second. Find, in seconds, how long it takes light from the sun to reach the earth.
DS/GRB/JMH: June 19, 2017
Page 9 of 19
* 54. The earth travels 958 million kilometres in its orbit around the sun each year (365 days). By
rounding off each number correct to 1 significant figure calculate how far the earth travels in
1 hour.
* 55. Repeat question 54 only use a calculator to do the actual calculation. (Round off your answer
correct to 2 significant figures.)
* 56. Use a calculator to help you find how many days there are in 1 million seconds. (Round off your
answer correct to 3 significant figures.)
EXERCISE 11: ( Number Patterns )
ORAL EXAMPLES: State the next 3 numbers in these sequences.
1. 1, 3, 5, 7,
,
,
2. 2, 4, 6, 8,
,
3. 3, 6, 9, 12,
,
,
4. 5, 10, 15, 20,
,
5. 1, 4, 7, 10,
,
,
6. 2, 6, 10, 14,
,
7. 1, 2, 4, 8,
,
,
8. 4, 40, 400, 4000,
9. 1, 2, 4, 7,
,
,
10. 1, 4, 9, 16,
,
WORKED EXAMPLE:
,
,
,
,
,
,
Check that the 4 given statements are true and then write down the next 2
statements as well as the 10th statement.
Checking
12 - 02 = 1 + 0
12 - 02 = 1 - 0 = 1 = 1 + 0
(This
can
be
22 - 12 = 2 + 1
22 - 12 = 4 - 1 = 3 = 2 + 0
done
mentally.)
32 - 22 = 3 + 2
32 - 22 = 9 - 4 = 5 = 3 + 2
42 - 32 = 4 + 3
42 - 32 = 16 - 9 = 7 = 4 + 3
By observing the pattern the next 2 statements are
52 - 42 = 5 + 4
62 - 52 = 6 + 5
The 10th statement is 102 - 92 = 10 + 9
WRITTEN EXAMPLES:
1.
3.
5.
13
23
33
43
+1=
+1=
+1=
+1=
Copy down and check that the 4 given statements are true.
Then write down the next 2 statements as well as the 10th
statement.
(1 + 1)(12 - 1 + 1)
2. 13 - 1 = (1 - 1)(12 + 1 + 1)
(2 + 1)(22 - 2 + 1)
23 - 1 = (2 - 1)(22 + 2 + 1)
(3 + 1)(32 - 3 + 1)
33 - 1 = (3 - 1)(32 + 3 + 1)
(4 + 1)(42 - 4 + 1)
43 - 1 = (4 - 1)(42 + 4 + 1)
22
32
42
52
12
22
32
42
=
=
=
=
1
1+3
1+3+5
1+3+5+7
+2x1+1
+2x2+1
+2x3+1
+2x4+1
= 12
= 22
= 32
= 42
4.
6.
12
22
32
42
=
=
=
=
22
32
42
52
-2x2+1
-2x3+1
-2x4+1
-2x5+1
2
2+4
2+4+6
2+4+6+8
=1x2
=2x3
=3x4
=4x5
DS/GRB/JMH: June 19, 2017
Page 10 of 19
1´ 2 ´ 3
3
2 ´ 3´ 4
1´ 2 + 2 ´ 3 =
3
3´ 4 ´ 5
1´ 2 + 2 ´ 3 + 3 ´ 4 =
3
4´5´6
1´ 2 + 2 ´ 3 + 3 ´ 4 + 4 ´ 5 =
3
1´ 2
2
2´3
1+ 2 =
2
3´ 4
1+ 2 + 3 =
2
4´5
1+ 2 + 3 + 4 =
2
1=
7.
1´ 2 =
8.
1´ 2 ´ 3
6
2 ´ 3´ 5
12 + 2 2 =
6
3´ 4 ´ 7
12 + 2 2 + 32 =
6
4´5´9
12 + 2 2 + 32 + 4 2 =
6
*11.
12 + 2 2 + 2 2 = 32
12 =
9.
1 = 2 -1
10.
1+ 2 = 2 2 - 1
1+ 2 + 4 = 2 3 - 1
1+ 2 + 4 + 8 = 2 4 - 1
*12.
13 = 1
2 2 + 32 + 6 2 = 7 2
23 = 3 + 5
32 + 4 2 + 12 2 = 132
33 = 7 + 9 + 11
4 2 + 5 2 + 20 2 = 212
4 3 = 13 + 15 + 17 + 19
* EXERCISE 12:
WORKED EXAMPLES:
WRITTEN EXAMPLES:
1. 3 8
2.
5.
4
16
6.
9.
3
64
10.
(2 )
4 3
Note:
a. Since 32 = 9 , then 3 = 9 = 2 9 (read as the square root of 9)
b. Likewise 34 = 81, and so 3 = 4 81 (read as the fourth root of 81)
c. Hence 3 125 (read as cube root of 125) = 5 (because 5 3 = 125 ).
Find the value of
3
4
27
10 000
3.
3
7.
5
1000
32
4.
3
1 000 000
8.
6
64
64
= 2 4 ´ 2 4 ´ 2 4 = 212 ( = 2 4+3 )
Hence
3
212 = 2 4 ( = 212¸3 )
Copy down and complete these statements in a similar way.
11. (3
)=3
2 4
Hence
2
4
2
2
2
?
?´?
´3 ´3 ´3 = 3 (3 )
12. (5
)=5
3 2
38 = 3? ( 3?¸? )
3
Hence
3
?
?´?
´ 5 = 5 (5 )
56 = 5? = 5?¸?
Find the value of the following leaving your answers as powers of 2, 3, 5 or 7.
13.
(7 )
5 2
14.
(3 )
6 4
15.
(2 )
4 5
16.
(5 )
3 6
DS/GRB/JMH: June 19, 2017
Page 11 of 19
715
21. 2 6 ´ 2 3
18.
312
22. 2 6 ¸ 2 3
19.
25. 312 ´ 34
26. 312 ¸ 34
27.
29. 5 8 ´ 5 2
30. 5 8 ¸ 52
31.
33. 710 ´ 7 5
34. 710 ¸ 75
35.
17.
3
2 30
3
23. ( 2 6 )
5
(3 )
(5 )
(7 )
12 4
8 2
10 5
58
24. 3 2 6
20.
4
28.
4
58
32.
36.
312
5
710
DS/GRB/JMH: June 19, 2017
Page 12 of 19
REVISION EXERCISE
Find the value of
1. 16 - 8 + 2
2.
16 - (8 + 2)
3. 16 ÷ 8 x 2
4.
16 ÷ (8 x 2)
5. 16 - 8 x 2
6.
(16 - 8) x 2
7. 16 + 8 ÷ 2
8.
(16 + 8) ÷ 2
9. 8 x 102
11. 6400
10.
12.
802
(22 + 32 + 62)
13. 4 x 52
14. 5 x 42
15. 42 x 52
16.
40 000
17. Use the fact that 55 = 3125 to help you find the value of
a.
56
b.
54
18. Write these products and quotients as powers of 2, 3 or 5.
a.
25 x 23
b.
36 ÷ 32
c.
54 ´ 52
53
19. State the factors of 97 given that 97 is a prime number.
20. Copy down and complete the following statements.
a.
{factors of 90} = {1, 2, 3, 5, 6, 9, , , , , ,
b.
{prime factors of 90} = {
}
c.
90 (written as the product of its prime factors) =
}
21.
Copy down and complete the following statements choosing from , , or
a.
1
{prime numbers}
b.
{2}
{prime numbers}
c.
3
{prime numbers}
d.
{4}
{prime numbers}
22.
a.
23.
Simplify by cancelling factors. (The first step has been started for you).
35 x 24
36 x 48
x9x8x
a.
(= x 7 xx 6 x )
b.
42
32 x 54 (= x 8 x 9 x )
A city has a population of 604 800 to the nearest 100.
a.
To how many significant figures is the population given?
b.
State the population correct to 3 significant figures.
c.
State the population correct to 2 significant figures.
Round off each number correct to 1 significant figure and then calculate
19 x 31
61 x 79
a.
b.
58
38 x 22
Copy down and check that the 4 given statements are true. Then write down the next 2
statements as well as the 10th statement.
12 x 22
13 = 4
22 x 32
13 + 23 = 4
32 x 42
13 + 23 + 33 = 4
42 x 52
13 + 23 + 33 + 43 = 4
24.
25.
26.
State the HCF of 10 and 25
b.
.
State the LCM of 10 and 25
DS/GRB/JMH: June 19, 2017
Page 13 of 19
TEST EXERCISE
(40 marks)
In questions 1 – 16 find the value of
40 -10 + 2 =___________
1.
2.
40 - (10 + 2) =___________
40 ¸ (10 ´ 2) =___________
3.
40 ¸10 ´ 2 =___________
4.
5.
40 +10 ´ 2 =___________
6.
7.
40 -10 ¸ 2 =___________
8.
9.
10.
11.
34 =___________
8 ´ 52 =___________
13.
15.
9002 =___________
(100 - 36) =___________
14.
16.
17.
Write down these products and quotients as a single power of 5.
(i) 56 ´ 53 =___________
18.
12.
(40 +10) ´ 2 =___________
(40 -10) ¸ 2 =___________
43 =___________
2
(8 ´ 5) =___________
900 =___________
100 - 36 =___________
(ii) 56 ¸ 53 =___________
56 ´ 52
=___________
54
(iii)
Use this table to help you find this product and quotient.
(i) 16 807 ´ 343 =
72 = 49
73 = 343
7 4 = 2401
7 = 16 807
Answer: ____________
(ii) 117 649 ¸ 2401 =
5
Answer:_____________
76 = 117 649
7 7 = 823 543
78 = 5 764 801
19.
Complete these statements by writing in Î, Ï, Ì or Ë.
(i) 0 (i.e. the number zero) ______ N = {natural numbers}.
(ii) {0} (i.e. the set with element zero) _____ W = {whole numbers}.
20.
State (i) the H.C.F. of 20 and 50. Answer:_____________
(ii) the L.C.M. of 20 and 50. Answer:_____________
21.
(a) Given that 323 =17 ´19 find
(i) {factors of 323} = {_____________________}.
(ii) {prime factors of 323} = {_________________ }.
(b) Given that 324 =182 find (i) {factors of 324} = {_____________________}.
(ii) {prime factors of 324} = {_________________ }.
(iii) Also write 324 as the product of its prime factors.
Answer: 324 = ________________________________
22.
54 ´ 56
by cancelling factors.
48
54 ´ 56 ___´ 6 ´ 8 ´ ___
=
=
48
___´ ___
Evaluate
DS/GRB/JMH: June 19, 2017
Page 14 of 19
23.
Given that 5 460 000 is correct to the nearest thousand
(i) state the number of significant figures it is correct to. Answer:___________
(ii) write down the number correct to 2 significant figures. Answer:__________
(iii) write down the number correct to 1 significant figure. Answer:___________
24.
Round each number off correct to 1 significant figure and then calculate
58 ´ 43
=___________
79
25.
Complete the first 3 statements by writing in the missing number. (It is the same for each statement).
Complete the remaining 3 statements.
12 + 22 + 32 = 3 ´ 22 + ____
22 + 32 + 42 = 3 ´ 32 + ____
32 + 42 + 52 = 3 ´ 42 + ____
42 + _________ = __________
52 + _________ = __________
102 + ________ = __________
DS/GRB/JMH: June 19, 2017
Page 15 of 19
EXTRA EXERCISE
1) a) A student claims that the prime numbers less than 50 can be found as follows – write down the first 3
prime numbers (2 , 3 and 5); then add 2 (to get 7), add 4 (to get 11), add 2 (to get 13), add 4,
add 2, add 4 and so on; and finally cross out the 2 digit numbers ending in 5.
i) Write down the numbers less than 50 which these steps give the student.
ii) Are any prime numbers left out? (If so state which ones).
iii) Are any numbers which are not prime left in? (If so state which ones).
b) A more logical way of finding the prime numbers less than 50 would be to list the whole numbers
between 1 and 50 and then go through deleting all multiples of 2, 3, 5 and 7 (but ofcourse, leaving the
numbers 2, 3, 5 and 7 themselves).
i) Why is it unnecessary to delete the multiples of 4 and 6?
ii) Why is it unnecessary to delete the multiples of prime numbers greater than 7?
c) If you had a list of whole numbers between 1 and 200 write down the only numbers you need to
delete the multiples of to find the prime numbers
i) between 50 and 100.
ii) between 100 and 150.
iii) between 150 and 200.
2) a)
Write down the value of 51, 52, 53, 54.
What is the last digit in any power of 5?
Evaluate 61, 62, 63, 64.
What is the last digit in any power of 6?
Explain why any power of 6 minus 1 is divisible by 5.
Evaluate 41, 4 2, 4 3, 4 4 .
What is the number pattern for the last digit in successive powers of 4?
What is the last digit in 4100?
Explain why any odd power of 4 plus 1 is divisible by 5 and any even power of 4 minus 1 is
divisible by 5.
i) Evaluate 91, 9 2, 9 3, 9 4 .
ii) What is the number pattern for the last digit in successive powers of 9?
iii) What is the last digit in 9100?
iv) Explain why any odd power of 9 plus 1 is divisible by 10 and any even power of 9 minus 1 is
divisible by 10.
i) Evaluate 21, 2 2, 2 3, 2 4 , 2 5, 2 6.
ii) What is the number pattern for the last digit in successive powers of 2?
iii) What is the last digit in 2100?
Repeat (e) for powers of 3.
Repeat (e) for powers of 7.
Repeat (e) for powers of 8.
i)
ii)
b) i)
ii)
iii)
c) i)
ii)
iii)
iv)
d)
e)
f)
g)
h)
3) Note: In this question you are to add the digits of various numbers (e.g. for 324, 3 + 2 + 4 = 9) and, if
necessary, keep adding the digits until a single digit number is obtained( e.g. for 867, 8 + 6 + 7 = 21 and
for 21, 2 + 1 = 3).
a) i) Find the single digit number pattern obtained by adding the digits of the natural numbers.
ii) Hence write down the single digit number pattern for the multiples of 3.
iii) Also write down the single digit number pattern for the multiples of 9.
b) Copy down and complete these statements.
i) A number is divisible by 3 if its digits eventually add up to _____, _____ or ____.
ii) A number is divisible by 9 if its digits eventually add up to _____.
iii) All powers of 3 are divisible by 9 (except for _____) and so the digits in these powers of 3
eventually add up to ____. (Some powers of 3 are listed in Exercise 4).
DS/GRB/JMH: June 19, 2017
Page 16 of 19
c) i)
Find the single digit number pattern obtained by adding the digits in the powers of 2.
(See Exercise 4).
ii) Hence write down the single digit number pattern for the powers of 4.
iii) Also write down the single digit number pattern for the powers of 8.
d) Use your results in (c) to help show that
i) the odd powers of 2 plus 1 are divisible by 3 and the even powers of 2 minus 1 are
divisible by 3.
ii) all powers of 4 minus 1 are divisible by 3.
iii) the odd powers of 8 plus 1 are divisible by 9 and the even powers of 8 minus 1 are
divisible by 9.
e) Copy down and complete this statement.
i) A number is divisible by 2 if it is an _________ number.
ii) A number is divisible by 6 if it is divisible by both ________ and ________.
f) i) Find the single digit number pattern obtained by adding the digits in the powers of 5.
(See Exercise 4).
ii) Hence show that the odd powers of 5 plus 1 are divisible by 6 and the even powers of 5 minus 1
are divisible by 6.
4) a) i) Write the whole numbers 180 and 126 as a product of their prime factors.
ii) Hence state their H.C.F. and L.C.M. (leaving your answers as a product of prime factors.)
b) i) Write down (180 x 126) as a product of prime factors.
ii) Show that (their H.C.F. x their L.C.M.) comes to the same product of prime factors.
c) Consider any 2 whole numbers. Does their product equal the product of their H.C.F. and L.C.M.?
d) Consider any 3 whole numbers. Does their product equal the product of their H.C.F. and L.C.M.?
5) a) Show that the L.C.M. of the whole numbers from 1 to 10 inclusive is 2520.
b) By looking at its prime factors state
i) the smallest whole number which is not a factor of 2520.
ii) the smallest even number which is not a factor of 2520.
iii) the largest square number which is a factor of 2520.
iv) the largest odd number which is a factor of 2520.
c) Find the number less than 2520 which has all except one of the whole numbers from 1 to 10
inclusive as factors.
d) Find the number which has
i) a remainder of 1 when it is divided by any of the whole numbers from 2 to 10 inclusive.
ii) a remainder of 1 less than the divisor when it is divided by any of the whole numbers from 1 to
10 inclusive (i.e. it has a remainder 1 when divided by 2, a remainder of 2 when divided by 3, a
remainder of 3 when divided by 4 and so on to a remainder of 9 when divided by 10).
6) Note: There are 10 single digit numbers (0, 1, 2, 3, ……, 9). There are 90 two digit numbers (10, 11, 12,
13,……., 99). The number of 2 digit numbers could have been calculated as follows: there are 9 ways of
choosing the first digit (1, 2, 3, …., 9) and for each way of choosing the first digit there are 10 ways of
choosing the second digit (0, 1, 2, 3, ……, 9) and so there are 9 x 10 different ways of choosing 2 digits.
This method of calculating the number of choices will be referred to as the multiplication principle.
a) Use the multiplication principle to find (in the form of 9 ´10n ) the number of
i) 3 digit numbers.
ii) 4 digit numbers.
iii) 5 digit numbers .
b) Use your answers in (a) to help you state (in the form of 9 ´10n ) the number of
i) 6 digit numbers.
ii) 10 digit numbers.
iii) n digit numbers.
DS/GRB/JMH: June 19, 2017
Page 17 of 19
c) Use the multiplication principle to find, as a product of prime factors, the number of ways of
choosing 3 cards from a pack of 52 cards if
i) each card is put back into the pack.
ii) each card is not put back into the pack.
d) Use the multiplication principle to find, as a product of prime factors, the number of car number
plates which can be made by using
i) 2 letters and 4 numbers.
ii) 3 letters and 3 numbers .
7) a) Copy down and complete this table for powers of 2.
Powers of 2
Factors for each power of 2 Number of factors
1
2 (= 2)
1, 2
2
22 (= 2 x 2)
1, 2, 2 x 2
23 (= 2 x 2 x 2)
24
25
b) Use the pattern of the results in the table above to state the number of factors in
i)
26
ii) 27
iii) 210
iv) 2 n (n Î N )
c) Would the number of factors in the table above be the same if the powers of 2 were replaced by
i) powers of 3, 5, 7, 11 etc (i.e. any prime number)?
ii) powers of 4, 6, 8, 9 etc (i.e. any number which is not prime)?
d) State the number of factors in
2
i)
3
ii)
iii) 33
iv) 34
3
v) 5
vi) 5 3
vii) 72
viii) 74
e) Write each of the following as a power of a prime number and hence state the number of factors in
3
3
i)
4 (= 2 )
ii) 8 (= 2 )
iii) 16
iv) 32
v) 9
vi) 27
vii) 25
viii) 49
f)
Copy down and complete this statement:
The number of factors in an is ________ provided that a is a __________ number.
g) i) How many factors does 2 3 ´ 32 have?
Hint: Each factor of 2 3 (ie 1, 2, 2 2 and 2 3) multiplied by each factor of 32 (i.e. 1, 3 and 32)
gives a different factor of 2 3 x 32.
ii) Does the number of factors in 2 3 x 32 equal the number of factors in 2 3 x the number of factors
in 32?
iii) How many factors does ( 2 3 x 32) x 5 have?
Hint: Each factor of 2 3 x 32 (the number was found in (ii) above) multiplied by each factor of 5
(there are 2 of them) gives a different factor of 2 3 x 32 x 5.
iv) Does the number of factors of 2 3 x 32 x 5 equal the number of factors in 2 3 x the number of
factors in 32 x the number of factors in 5 ?
h) Find the number of factors in these products.
2
2
3
4
3x 5
i)
ii) 2 x 5 3
iii) 3 x 5
2
2
2
3
2
2
v) 2 x 3 x 5
vi) 2 x 3 x 5
vii) 2 x 3 x 5
i)
j)
4
3
iv) 2 x 5
3
viii) 2 x 3 x 5 2
Copy down and complete this statement:
The number of factors in am x bn is ______ provided that a and b are different ______ numbers.
Write each of the following numbers as a product of their prime factors. Hence find how many
factors each number has.
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i) 100
v) 60
ii)
vi)
200
120
iii) 1000
vii) 180
iv) 2000
viii) 360
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