## Chapter 3 – HCF and LCM

## Practice Set 10

**1. Which number is neither a prime number nor a composite number?****Ans:** 1 is neither a prime number nor a composite number.

**2. Which of the following are pairs of co-primes?****(i) 8, 14****Solution:**

Factors of 8: 1, 2, 4, 8

Factors of 14: 1, 2, 7, 14

Common factors of 8 and 14: 1, 2

∴ 8 and 14 are not a pair of co-prime numbers.

**(ii) 4, 5 ****Solution:**

Factors of 4: 1, 4

Factors of 5: 1, 5

Common factors of 4 and 5: 1

∴ 4 and 5 are a pair of co-prime numbers.

**(iii) 17, 19 ****Solution:**

Factors of 17: 1, 17

Factors of 19: 1, 19

Common factors of 17 and 19: 1

∴ 17 and 19 are a pair of co-prime numbers.

**(iv) 27, 15****Solution:**

Factors of 27: 1, 3, 9, 27

Factors of 15: 1, 3, 5, 15

Common factors of 27 and 15 : 1, 3

∴ 27 and 15 are not a pair of co-prime numbers.

**3. List the prime numbers from 25 to 100 and say how many they are.****Ans:** 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

There are altogether 16 prime numbers from 25 to 100.

**4. Write all the twin prime numbers from 51 to 100.****Ans:**

(i) 59 and 61

(ii) 71 and 73

**5. Write 5 pairs of twin prime numbers from 1 to 50.****Ans:**

(i) 3, 5

(ii) 5, 7

(iii) 11, 13

(iv) 17, 19

(v) 29, 31

(vi) 41, 43

**6. Which are the even prime numbers?****Ans:** 2 is the only even prime number.

## Practice set 11

**Factorise the following numbers into primes.**

**(i) 32 **

**Solution:**

∴ 32 = 2 × 2 × 2 × 2 × 2

**(ii) 57 **

**Solution:**

∴ 57 = 3 × 19

**(iii) 23 **

**Solution:**

∴ 23 = 23 × 1

**(iv) 150 **

**Solution:**

**∴ 150 = **2 × 3 × 5 × 5

**(v) 216**

**Solution:**

∴ 216 = 2 × 2 × 2 × 3 × 3 × 3

**(vi) 208 **

**Solution:**

∴ 208 = 2 × 2 × 2 × 2 × 13

**(vii) 765 **

**Solution:**

∴ 765 = 3 × 3 × 5 × 17

**(viii) 342 **

**Solution:**

∴ 342 = 2 × 3 × 3 × 19

**(ix) 377 **

**Solution:**

∴ 377 = 13 × 29

**(x) 559**

**Solution:**

∴ 559 = 13 × 43

## Practice set 12

**1. Find the HCF.****(i) 25, 40 ****Solution:**

25 = 5 × 5

40 = 2 × 2 × 2 × 5

∴ HCF of 25 and 40 = 5

**(ii) 56, 32 **

**Solution:**

56 = 2 × 2 × 2 × 7

32 = 2 × 2 × 2 × 2 × 2

∴ HCF of 56 and 32 = 2 × 2 × 2

∴ HCF of 56 and 32 = 8

**(iii) 40, 60, 75 **

**Solution:**

40 = 2 × 2 × 2 × 5

60 = 2 × 2 × 3 × 5

75 = 3 × 5 × 5

∴ HCF of 40, 60 and 75 = 5

**(iv) 16, 27**

**Solution:**

16 = 2 × 2 × 2 × 2

27 = 3 × 3 × 3

∴ HCF of 16 and 27 = 1

**(v) 18, 32, 48 **

**Solution:**

18 = 2 × 3 × 3

32 = 2 × 2 × 2 × 2 × 2

48 = 2 × 2 × 2 × 2 × 3

∴ HCF of 18, 32 and 48 = 2

**(vi) 105, 154 **

**Solution:**

777 = 3 × 7 × 37

315 = 3 × 3 × 5 × 7

588 = 2 × 2 × 3 × 7 × 7

∴ HCF of 777, 315 and 588 = 3 × 7

∴ HCF of 777, 315 and 588 = 21

**2. Find the HCF by the division method and reduce to the simplest form.**

**(i) \(\large \frac {275}{525}\)**

**Solution:**

HCF of 275 and 525 = 25

∴ \(\large \frac {275}{525}\) = \(\large \frac {275\,÷\,25}{525\,÷\,25}\) = \(\large \frac {11}{21}\)

**(ii) \(\large \frac {76}{133}\)**

**Solution:**

HCF of 76 and 133 = 19

∴ \(\large \frac {76}{133}\) = \(\large \frac {76\,÷\,19}{133\,÷\,19}\) = \(\large \frac {4}{7}\)

**(iii) \(\large \frac {161}{69}\)**

**Solution:**

HCF of 161 and 69 = 23

∴ \(\large \frac {161}{69}\) = \(\large \frac {161\,÷\,23}{69\,÷\,23}\) = \(\large \frac {7}{3}\)

## Practice set 13

**1. Find the LCM.****(i) 12, 15 ****Solution:**

12 = 2 × 2 × 3

15 = 3 × 5

∴ LCM of 12 and 15 = 2 × 2 × 3 × 5

∴ LCM of 12 and 15 = 60

**(ii) 6, 8, 10 **

**Solution:**

6 = 2 × 3

8 = 2 × 2 × 2

10 = 2 × 5

∴ LCM of 6, 8 and 10 = 2 × 3 × 2 × 2 × 5

∴ LCM of 6, 8 and 10 = 120

**(iii) 18, 32 **

**Solution:**

18 = 2 × 3 × 3

32 = 2 × 2 × 2 × 2 × 2

∴ LCM of 18 and 32 = 2 × 3 × 3 × 2 × 2 × 2 × 2

∴ LCM of 18 and 32 = 288

**(iv) 10, 15, 20 **

**Solution:**

10 = 2 × 5

15 = 3 × 5

20 = 2 × 2 × 5

∴ LCM of 10, 15 and 20 = 2 × 5 × 3 × 2

∴ LCM of 10, 15 and 20 = 60

**(v) 45, 86**

**Solution:**

45 = 5 × 3 × 3

86 = 2 × 43

∴ LCM of 45 and 86 = 5 × 3 × 3 × 2 × 43

∴ LCM of 45 and 86 = 3870

**(vi) 15, 30, 90 **

**Solution:**

15 = 3 × 5

30 = 2 × 3 × 5

90 = 2 × 5 × 3 × 3

∴ LCM of 15, 30 and 90 = 3 × 5 × 2 × 3

∴ LCM of 15, 30 and 90 = 90

**(vii) 105, 195 **

**Solution:**

105 = 3 × 5 × 7

195 = 3 × 5 × 13

∴ LCM of 105 and 95 = 3 × 5 × 7 × 13

∴ LCM of 105 and 95 = 1365

**(viii) 12, 15, 45 **

**Solution:**

12 = 2 × 2 × 3

15 = 3 × 5

45 = 5 × 3 × 3

∴ LCM of 12, 15 and 45 = 3 × 5 × 2 × 2 × 3

∴ LCM of 12, 15 and 45 = 180

**(ix) 63, 81**

**Solution:**

63 = 3 × 3 × 7

81 = 3 × 3 × 3 × 3

∴ LCM of 63 and 81 = 3 × 3 × 7 × 3 × 3

∴ LCM of 63 and 81 = 567

**(x) 18, 36, 27**

**Solution:**

18 = 2 × 3 × 3

36 = 2 × 2 × 3 × 3

27 = 3 × 3 × 3

∴ LCM of 18, 36 and 27 = 2 × 3 × 3 × 2 × 3

∴ LCM of 18, 36 and 27 = 108

**2. Find the HCF and LCM of the numbers given below. Verify that their product is equal to the product of the given numbers.****(i) 32, 37 ****Solution:**

32 = 2 × 2 × 2 × 2 × 2

37 = 37

HCF of 32 and 37 = 1

LCM of 32 and 37 = 2 × 2 × 2 × 2 × 2 × 37

∴ LCM of 32 and 37 = 1184

HCF × LCM = 1 × 1184

∴ HCF × LCM = 1184

Product of the given numbers = 32 × 37

∴ Product of the given numbers = 1184

∴ HCF × LCM = Product of the given numbers

**Hence verified.**

**(ii) 46, 51 **

**Solution:**

46 = 2 × 23

51 = 3 × 17

HCF of 46 and 51 = 1

LCM of 46 and 51 = 2 × 23 × 3 × 17

∴ LCM of 46 and 51 = 2346

HCF × LCM = 1 × 2346

∴ HCF × LCM = 2346

Product of the given numbers = 46 × 51

∴ Product of the given numbers = 2346

∴ HCF × LCM = Product of the given numbers

**Hence verified.**

**(iii) 15, 60 **

**Solution:**

15 = 3 × 5

60 = 2 × 2 × 3 × 5

HCF of 15 and 60 = 3 × 5

∴ HCF of 15 and 60 = 15

LCM of 15 and 60 = 3 × 5 × 2 × 2

∴ LCM of 15 and 60 = 60

HCF × LCM = 15 × 60

∴ HCF × LCM = 900

Product of the given numbers = 15 × 60

∴ Product of the given numbers = 900

∴ HCF × LCM = Product of the given numbers

**Hence verified.**

**(iv) 18, 63 **

**Solution:**

18 = 2 × 3 × 3

63 = 3 × 3 × 7

HCF of 18 and 63 = 3 × 3

∴ HCF of 18 and 63 = 9

LCM of 18 and 63 = 3 × 3 × 2 × 7

∴ LCM of 18 and 63 = 126

HCF × LCM = 9 × 126

∴ HCF × LCM = 1134

Product of the given numbers = 18 × 63

∴ Product of the given numbers = 1134

∴ HCF × LCM = Product of the given numbers

**Hence verified.**

**(v) 78, 104**

**Solution:**

78 = 2 × 3 × 13

104 = 2 × 2 × 2 × 13

HCF of 78 and 104 = 2 × 13

∴ HCF of 78 and 104 = 26

LCM of 78 and 104 = 2 × 13 × 3 × 2 × 2

∴ LCM of 78 and 104 = 312

HCF × LCM = 26 × 312

∴ HCF × LCM = 8112

Product of the given numbers = 78 × 104

∴ Product of the given numbers = 8112

∴ HCF × LCM = Product of the given numbers

**Hence verified.**

## Practice set 14

**1. Choose the right option.**

**(i) The HCF of 120 and 150 is ________.**

**(1) 30 **

**(2) 45 **

**(3) 20 **

**(4) 120**

**Ans:** Option (1) – 30

**Solution:**

120 = 2 × 2 × 2 × 3 × 5

150 = 2 × 3 × 5 × 5

HCF of 120 and 150 = 2 × 3 × 5

∴ HCF of 120 and 150 = **30**

**(ii) The HCF of this pair of numbers is not 1.**

**(1) 13, 17 **

**(2) 29, 20 **

**(3) 40, 20 **

**(4) 14, 15**

**Ans:** Option (3) – 40, 20

**Solution: **

40 = 2 × 2 × 2 × 5

60 = 2 × 2 × 5

HCF of 120 and 150 = 2 × 2 × 5

∴ HCF of 120 and 150 = 20 **≠ 1**

**2. Find the HCF and LCM.****(i) 14, 28****Solution:**

14 = 2 × 7

28 = 2 × 2 × 7

HCF of 14 and 28 = 2 × 7

∴ HCF of 14 and 28 = 14

LCM of 14 and 28 = 2 × 7 × 2

∴ LCM of 14 and 28 = 28

**(ii) 32, 16 **

**Solution:**

32 = 2 × 2 × 2 × 2 × 2

16 = 2 × 2 × 2 × 2

HCF of 32 and 16 = 2 × 2 × 2 × 2

∴ HCF of 32 and 16 = 16

LCM of 32 and 16 = 2 × 2 × 2 × 2 × 2

∴ LCM of 32 and 16 = 32

**(iii) 17, 102, 170 **

**Solution:**

17 = 17

102 = 2 × 3 × 17

170 = 2 × 5 × 17

HCF of 17, 102 and 170 = 17

LCM of 17, 102 and 170 = 17 × 2 × 3 × 5

∴ LCM of 17, 102 and 170 = 510

**(iv) 23, 69 **

**Solution:**

23 = 23

69 = 3 × 23

HCF of 23 and 69 = 23

LCM of 23 and 69 = 2 × 23

∴ LCM of 23 and 69 = 69

**(v) 21, 49, 84**

**Solution:**

21 = 3 × 7

49 = 7 × 7

84 = 2 × 2 × 3 × 7

HCF of 21, 49 and 84 = 7

LCM of 21, 49 and 84 = 7 × 3 × 7 × 2 × 2

∴ LCM of 21, 49 and 84 = 588

**3. Find the LCM.****(i) 36, 42 ****Solution:**

36 = 2 × 2 × 3 × 3

42 = 2 × 3 × 7

∴ LCM of 36 and 42 = 2 × 3 × 2 × 3 × 7

∴ LCM of 36 and 42 = 252

**(ii) 15, 25, 30 ****Solution:**

15 = 3 × 5

25 = 5 × 5

30 = 2 × 3 × 5

∴ LCM of 15, 25 and 30 = 3 × 5 × 5 × 2

∴ LCM of 15, 25 and 30 = 150

**(iii) 18, 42, 48 ****Solution:**

18 = 2 × 3 × 3

42 = 2 × 3 × 7

48 = 2 × 2 × 2 × 2 × 3

∴ LCM of 18, 42 and 48 = 2 × 3 × 3 × 7 × 2 × 2 × 2

∴ LCM of 18, 42 and 48 = 1008

**(iv) 4, 12, 20 **

**Solution:**

4 = 2 × 2

12 = 2 × 2 × 3

20 = 2 × 2 × 5

∴ LCM of 4, 12 and 20 = 2 × 2 × 3 × 5

∴ LCM of 18, 42 and 48 = 60

**(v) 24, 40, 80, 120****Solution:**

24 = 2 × 2 × 2 × 3

40 = 2 × 2 × 2 × 5

80 = 2 × 2 × 2 × 2 × 5

120 = 2 × 2 × 2 × 3 × 5

∴ LCM of 24, 40, 80 and 120 = 2 × 2 × 2 × 3 × 5 × 2

∴ LCM of 24, 40, 80 and 120 = 240

**4. Find the smallest number which when divided by 8, 9, 10, 15, 20 gives a remainder of 5 every time.****Solution:**

The smallest number for division will be the LCM of 8, 9, 10,15 and 20.

8 = 2 × 2 × 2

9 = 3 × 3

10 = 2 × 5

15 = 3 × 5

20 = 2 × 2 × 5

∴ LCM of 8, 9, 10,15 and 20 = 2 × 2 × 2 × 3 × 3 × 5

∴ LCM of 8, 9, 10,15 and 20 = 360

∴ Smallest number = LCM + Remainder

∴ Smallest number = 360 + 5

∴ Smallest number = 365

**Ans:** The required smallest number is 365.

**5. Reduce the fractions ****\(\large \frac {348}{319}\), \(\large \frac {221}{247}\), \(\large \frac {437}{551}\) to the lowest terms.****Solution:**

HCF of 348 and 319 = 29

\(\large \frac {348}{319}\) = \(\large \frac {348\,÷\,29}{319\,÷\,29}\) = \(\large \frac {12}{11}\)

HCF of 221 and 247 = 13

∴ \(\large \frac {221}{247}\) = \(\large \frac {221\,÷\,13}{247\,÷\,13}\) = \(\large \frac {17}{19}\)

HCF of 437 and 551 = 19

∴ \(\large \frac {437}{551}\) = \(\large \frac {437\,÷\,19}{551\,÷\,19}\) = \(\large \frac {23}{29}\)