[3 marks sum]

Question 13 Marks

Find the smallest number that is completely divisible by 28 and 42.

Answer

We know that the least number which is divisible by 28 and 42 is their L.C.M.

2	28,	42
7	14,	21
	2,	3

L.C.M. of 28 and 42 = 2 × 2 × 3 × 7 = 84

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Question 23 Marks

The H.C.F. and the L.C.M. of two numbers are $50$ and $300$ respectively. If one of the numbers is $150,$ find the other one.

Answer

$\text { H.C.F. }=50 $
$\text { L.C.M. }=300$
Product of L.C.M. and H.C.F. $=300 \times 50=15000$
One number $=150$
The other number $=\frac{\text { Product of L.C.M and H.C.M }}{\text { One number }}=\frac{15000}{150}=100$

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Question 33 Marks

Using the common multiple method, find the L.C.M. of the following:
3, 6, 9 and 12

Answer

3, 6, 9 and 12

3	3,	6,	9,	12
2	1,	2,	3,	4
	1,	1,	3,	2

∴ L.C.M. = 3 x 2 x 3 x 2 = 36

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Question 43 Marks

Using the common multiple method, find the L.C.M. of the following:
10, 15 and 20

Answer

10, 15 and 20

2	10,	15	20
2	5,	15	10
5	5,	15,	5
	1,	3,	1

∴ L.C.M. = 2 x 2 x 5 x 3 = 60

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Question 53 Marks

Using the common multiple method, find the L.C.M. of the following:
8, 12 and 24

Answer

8, 12 and 24

4	8,	12,	24
3	2,	3,	6
2	2,	1,	2
	1,	1,	1

∴ L.C.M. = 4 x 3 x 2 = 24

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Question 63 Marks

Show that 45 and 56 are co-prime numbers.

Answer

The HCF of two co-prime numbers is always HCF of 45 and 56

From above it is proved that HCF of 45 and 56 is 1
Hence 45 and 56 are co-prime numbers.

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Question 73 Marks

Use a method of your own choice to find the H.C.F. of 30, 60, 90 and 105.

Answer

Factor of $60=F_{60}=2 \times 2 \times 3 \times 5$
Factor of $90=F_{90}=2 \times 3 \times 3 \times 5$
and factor of $105= F _{105}=3 \times 5 \times 7$
Now the common factor of $30,60,90$ and $105=3$ and
5 H.C.F. $=3 \times 5=15$

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Question 83 Marks

Use a method of your own choice to find the H.C.F. of 24, 36, 60, and 132.

Answer

Factor of $24=F_{24}=2 \times 2 \times 2 \times 3$
Factor of $36=F_{36}=2 \times 2 \times 3 \times 3$
Factor of $60=F_{60}=2 \times 2 \times 3 \times 5$
and Factor of $132=F_{132}=2 \times 2 \times 3 \times 11$
Now the common factors of $24,36,60$ and $132=2,2$ and 3
H.C.F. $=2 \times 2 \times 3=12$

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Question 93 Marks

Use a method of your own choice to find the H.C.F. of 66, 33 and 132.

Answer

Factor of $66=F_{66}=2 \times 3 \times 11$
Factor of $33=F_{33}=3 \times 11$
and factor of $132=F_{132}=2 \times 2 \times 3 \times 11$
Now the common factor of 66,33 and $132=3$ and 11
H.C.F. $=3 \times 11=33$

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Question 103 Marks

Use a method of your own choice to find the H.C.F. of 48, 36 and 96.

Answer

Solution Factor of $48=F_{48}=2 \times 2 \times 2 \times 2 \times 3$
Factor of $36=F_{36}=2 \times 2 \times 3 \times 3$
and factor of $96=F_{96}=2 \times 2 \times 2 \times 2 \times 2 \times 3$
Now the common factor of 48,36 and $96=2,2$ and 3
H.C.F. $=2 \times 2 \times 3=12$

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Question 113 Marks

Use a method of your own choice to find the H.C.F. of 45, 75 and 135.

Answer

Factor of $45=F_{45}=3 \times 3 \times 5$
Factor of $75= F _{75}=3 \times 5 \times 5$
and Factor of $135=F_{135}=3 \times 3 \times 3 \times 5$
Now the common factors of 45,75 and $135=3$ and 5
H.C.F. $=3 \times 5=15$

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Question 123 Marks

Using the division method, find the H.C.F. of the following:
70,80,120 and 150

Answer

70,80,120 and 150

Since last division is 10
$\therefore$ H.C.F. of $70,80,120$ and $150=10$

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Question 133 Marks

Using the division method, find the H.C.F. of the following:
7, 14 and 24

Answer

7, 14 and 24

Since last division is 1
$\therefore$ H.C.F. of 7,14 and $24=1$

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Question 143 Marks

Using the division method, find the H.C.F. of the following:
18 and 30

Answer

18 and 30

Since last division is 6
$\therefore$ H.C.F. of 18 and $30=6$

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Question 153 Marks

Using the division method, find the H.C.F. of the following:
16 and 24

Answer

16 and 24

Since last division is 8
$\therefore$ H.C.F. of 16 and $24=8$

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Question 163 Marks

Using the prime factor method, find the H.C.F. of 12, 16 and 28

Answer

Prime factor of 12 = 2 x 2 x 3
Prime factor of 16 = 2 x 2 x 2 x 2
Prime factor of 28 = 2 x 2 x 7
Common prime factor between 12, 16 and 28 = 2 x 2
H.C.F. of 12, 16 and 28 = 2 x 2 = 4

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Question 173 Marks

Using the prime factor method, find the H.C.F. of 48, 84 and 88

Answer

Prime factor of 48 = 2 x 2 x 2 x 2 x 3
Prime factor of 84 = 2 x 2 x 3 x 7
Prime factor of 88 = 2 x 2 x 2 x 11
Common prime factor of 48, 84 and 88 = 2 x 2
H.C.F. of 48, 84 and 88 = 2 x 2 = 4

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Question 183 Marks

Using the prime factor method, find the H.C.F. of 40, 60 and 80

Answer

Prime factor of 40 = 2 x 2 x 2 x 5
Prime factor of 60 = 2 x 2 x 3 x 5
Prime factor of 80 = 2 x 2 x 2 x 2 x 5
Common prime factor = 2 x 2 x 5
H.C.F. of 40, 60 and 80 = 2 x 2 x 5 = 20

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Question 193 Marks

Using the prime factor method, find the H.C.F. of 24 and 49

Answer

Prime factor of 24 = 2 x 2 x 2 x 3
Prime factor of 49 = 7 x 7
No common prime factor, the number are co-prime.
H.C.F. of 24 and 49 = 1.

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Question 203 Marks

Using the prime factor method, find the H.C.F. of 5 and 8

Answer

Prime factor of 5 = 5
Prime factor of 8 = 2 x 2 x 2
No common prime factor
H.C.F. of 5 and 8 = 1
(as both the number are co-prime)

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Question 213 Marks

Using the common factor method, find the H.C.F. of 24, 36, 45 and 60

Answer

$F_{24}=1,2,3,4,6,8,12,24$
$F_{36}=1,2,3,4,6,12,18,36$
$F_{45}=1,3,5,9,15,45$
$F_{60}=1,3,4,5,6,10,12,15,20,30,60$
Common factor between $24,36,45$ and $60=1,3$ H.C.F. of $24,36,45$ and $60=3$

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Question 223 Marks

Using the common factor method, find the H.C.F. of 8, 12 and 18

Answer

$F_8=1,2,4,8 $
$F_{12}=1,2,3,4,6,12 $
$F_{18}=1,2,3,6,9,18$
Common factors between 8,12 and $18=1,2$
H.C.F. of 8,12 and $18=2$

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Question 233 Marks

Write the prime factors of 27

Answer

Prime factors of 27 = 3