**1. Soha has got three numbers 6, 72 and 120. Find the HCF and LCM of the numbers using the prime factorisation method.**

a) 6 and 360

b) 12 and 240

c) 15 and 180

d) 18 and 160

**Answer:** a) 6 and 360

**Explanation:** Prime factorisation of 6 = 2 × 3 = 2^{1} × 3^{1}

Prime factorisation of 72 = 2 × 2 × 2 × 3 × 3 = 2^{3} × 3^{2}

Prime factorisation of 120 = 2 × 2 × 2 × 3 × 5 = 2^{3} × 3 x 5

HCF = 2^{1} × 3^{1} = 6

LCM = 2^{3} × 3^{2} x 5 = 360

**2. Find the HCF and LCM of 15 and 20.**

a) 5 and 20

b) 5 and 60

c) 10 and 20

d) 10 and 60

**Answer:** b) 5 and 60

**Explanation:** HCF- 15 = 3^{1} x 5^{1}

20 = 2 x 2 x 5 = 2^{2} x 5^{1}

The highest common factor of 15 and 20 is 5.

LCM- (15, 20) = (3 x 5) x (22) = 3 x 22 x 5 = 60

Therefore, HCF = 5 and LCM = 60.

**3. The HCF of two numbers is 15 and their LCM is 420. If one of the numbers is 60, find the other number.**

a) 8

b) 65

c) 98

d) 105

**Answer:** d) 105

**Explanation:** Let the other number be ‘y’.

HCF × LCM = Product of numbers

15 x 420 = 60 x y

(15 x 420)/60=y

15 x 7 = y

y = 105

**4. Find the HCF of 12 and 18 using the division method.**

a) 3

b) 4

c) 6

d) 8

**Answer:** c) 6

**Explanation:** Step 1: Treat the smallest number i.e., 12 as divisor and the bigger number i.e., 18 as a dividend.

Step 2: The remainder 6 becomes the divisor and the divisor 12 becomes the dividend.

Step 3: Repeat this process till the remainder becomes zero. The last divisor is the H.C.F.

**5. John wants to build a fence around his rectangular garden. The length of the garden is 18 m and the width is 12 m. He wants to divide the garden into identical square plots. What is the largest possible size of each square plot?**

a) 3 m x3 m

b) 4 m x 4 m

c) 5 m x 5 m

d) 6 m x 6 m

**Answer:** d) 6 m x 6 m

**Explanation:** The area of the garden is 18 x 12 = 216 square meters.

HCF (18, 12) = 6

To divide the garden into identical square plots, the size of each square plot should be a factor of 6.

LCM (18, 12) = 36

Therefore, the largest possible size of each square plot is 6 m by 6 m, and there will be a total of 6 x 6 = 36 square plots in the garden.

**1. The ages of three friends are 18 years, 24 years, and 30 years. Find the HCF of their ages.**

a) 3 years

b) 4 years

c) 5 years

d) 6 years

**Answer:** d) 6 years

**2. A farmer has three fields with areas 36 acres, 48 acres and 60 acres. What is the largest possible square field he can create by dividing these fields equally?**

a) 124 acres

b) 132 acres

c) 136 acres

d) 144 acres

**Answer:** d) 144 acres

**3. Three students need to assemble gift bags with an equal number of pencils. If one has 8 pencils, another has 9 pencils and the third has 25 pencils, what is the greatest number of pencils they can have in each bag?**

a) 1 pencil

b) 2 pencil

c) 3 pencil

d) 4 pencil

**Answer:** a) 1 pencil

**4. A factory produces bolts in batches of 12, 15 and 20. What is the minimum number of bolts they can produce to fill complete batches of each size?**

a) 60

b) 68

c) 80

d) 92

**Answer:** a) 60

**5. Three buses depart from a station at intervals of 8 minutes, 12 minutes, and 18 minutes. After how many minutes will they all depart simultaneously again?**

a) 64 minutes

b) 72 minutes

c) 80 minutes

d) 92 minutes

**Answer:** b) 72 minutes

**6. The HCF of two numbers is 4 and the LCM is 48. If one number is 16, find the other number.**

a) 8

b) 10

c) 12

d) 14

**Answer:** c) 12

**7. What is the largest number that, when divided into 1657 and 2037, leaves a remainder of 6 and 5, respectively?**

a) 88

b) 108

c) 127

d) 140

**Answer:** c) 127

**8. What is the smallest number that, when divided by 20, 25, 35 and 40, leaves remainders of 14, 19, 29 and 34, respectively?**

a) 1045

b) 1145

c) 1258

d) 1394

**Answer:** d) 1394

**9. If the highest common factor (HCF) of two numbers is 11 and their least common multiple (LCM) is 7700 and one of the numbers is 275, what is the value of the other number?**

a) 215

b) 245

c) 258

d) 308

**Answer:** d) 308

**10. The LCM (Least Common Multiple) of two numbers is 45 times their HCF (Highest Common Factor). If one of the numbers is 125 and the sum of the HCF and LCM is 1150, what is the value of the other number?**

a) 205

b) 225

c) 258

d) 328

**Answer:** b) 225

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