Question 1

Find the LCM of 12, 18, and 24.

Accepted Answer

Step 1: Prime factorize each number. 12 = 2² × 3, 18 = 2 × 3², 24 = 2³ × 3. Step 2: LCM = product of each prime factor with the highest power = 2³ × 3² = 8 × 9 = 72. Therefore, answer is 72.

Question 2

The HCF of two numbers is 12 and their LCM is 360. If one of the numbers is 72, what is the other number?

Accepted Answer

Step 1: Use the relation: Product of two numbers = HCF × LCM. Step 2: First number × Second number = 12 × 360 = 4320. Step 3: Second number = 4320 ÷ 72 = 60. Therefore, the other number is 60.

Question 3

Three bells ring at intervals of 18 minutes, 24 minutes, and 36 minutes respectively. If they all ring together at 9:00 AM, at what time will they ring together again for the third time?

Accepted Answer

Step 1: Find LCM(18, 24, 36). Prime factorisation: 18 = 2 × 3², 24 = 2³ × 3, 36 = 2² × 3². Step 2: LCM = 2³ × 3² = 8 × 9 = 72 minutes. Step 3: They ring together every 72 minutes. First time together: 9:00 AM (given). Second time: 9:00 AM + 72 min = 10:12 AM. Third time: 10:12 AM + 72 min = 11:24 AM. Therefore, the answer is 11:24 AM.

Question 4

The LCM of two numbers is 2520 and their HCF is 12. If one number is 168, find the other number.

Accepted Answer

Step 1: Use the fundamental relationship: For any two numbers a and b, a × b = LCM(a,b) × HCF(a,b). Step 2: Given LCM = 2520, HCF = 12, and one number = 168. Let the other number be x. Step 3: Therefore, 168 × x = 2520 × 12. Step 4: 168 × x = 30240. Step 5: x = 30240 ÷ 168 = 180. Step 6: Verification: HCF(168, 180) = 12 ✓ and LCM(168, 180) = 2520 ✓ (since 168 = 12 × 14 and 180 = 12 × 15, where 14 and 15 are coprime).

Question 5

The HCF of two numbers is 16 and their LCM is 480. How many pairs of numbers satisfy this condition?

Accepted Answer

Step 1: If HCF = 16, both numbers can be written as 16m and 16n where HCF(m,n) = 1 (coprime). Step 2: Then LCM = 16 × m × n = 480, so m × n = 30. Step 3: Find coprime pairs (m,n) whose product is 30. Prime factorisation of 30 = 2 × 3 × 5. Step 4: Coprime pairs: (1,30), (2,15), (3,10), (5,6). That is 4 pairs. Step 5: Each pair gives two ordered pairs, but the question asks for unordered pairs of numbers. Therefore, there are 4 pairs.

Question 6

The LCM of two numbers is 12 times their HCF. If the sum of the HCF and LCM is 403, find the HCF of the two numbers.

Accepted Answer

Step 1: Let HCF = h and LCM = l. Given: l = 12h and h + l = 403. Step 2: Substitute: h + 12h = 403, so 13h = 403. Step 3: h = 403 ÷ 13 = 31. Step 4: Verify: LCM = 12 × 31 = 372. Sum = 31 + 372 = 403 ✓. Therefore, HCF = 31.

Question 7

Two bells ring at intervals of 18 minutes and 24 minutes respectively. If they ring together at 9:00 AM, at what time will they ring together again?

Accepted Answer

Step 1: Find LCM of 18 and 24. Prime factorization: 18 = 2 × 3², 24 = 2³ × 3. Step 2: LCM = 2³ × 3² = 8 × 9 = 72 minutes. Step 3: They will ring together after 72 minutes. Step 4: 9:00 AM + 72 minutes = 10:12 AM. Therefore, the answer is 10:12 AM.

Question 8

The HCF of two numbers is 18. If the numbers are in the ratio 2:3, find the larger number.

Accepted Answer

Step 1: If HCF is 18 and numbers are in ratio 2:3, then the numbers are 2×18 and 3×18. Step 2: First number = 2 × 18 = 36. Step 3: Second number = 3 × 18 = 54. Step 4: The larger number is 54. Therefore, the answer is 54.

Question 9

The HCF of two numbers is 8. Which of the following cannot be their LCM?

Accepted Answer

Step 1: If HCF of two numbers is 8, then both numbers must be multiples of 8. Step 2: Therefore, their LCM must also be a multiple of 8. Step 3: Check each option: 64 = 8 × 8 ✓, 80 = 8 × 10 ✓, 96 = 8 × 12 ✓, 100 = 8 × 12.5 ✗ (not a multiple of 8). Step 4: 100 is not divisible by 8. Therefore, 100 cannot be their LCM.

Question 10

Three numbers are in the ratio 2:3:4. If their HCF is 5, find the largest number.

Accepted Answer

Step 1: Let the numbers be 2k, 3k, and 4k. Step 2: HCF(2k, 3k, 4k) = k × HCF(2, 3, 4) = k × 1 = k. Step 3: Since HCF = 5, we have k = 5. Step 4: The three numbers are 10, 15, and 20. Step 5: Largest number = 20.

Question 11

What is the LCM of 24, 36, and 48?

Accepted Answer

Step 1: Find prime factorization: 24 = 2³ × 3, 36 = 2² × 3², 48 = 2⁴ × 3. Step 2: LCM = highest power of each prime = 2⁴ × 3² = 16 × 9 = 144.

Question 12

Find the HCF of 56, 84, and 140 using the prime factorization method.

Accepted Answer

Step 1: Find prime factorization of each number. 56 = 2³ × 7, 84 = 2² × 3 × 7, 140 = 2² × 5 × 7. Step 2: HCF is the product of the lowest powers of all common prime factors. Step 3: Common prime factors are 2 and 7. Step 4: HCF = 2² × 7 = 4 × 7 = 28. Therefore, the answer is 28.

Question 13

Find the LCM of 24, 36, and 40.

Accepted Answer

Step 1: Find prime factorization of each number. 24 = 2³ × 3, 36 = 2² × 3², 40 = 2³ × 5. Step 2: LCM is the product of highest powers of all prime factors. Step 3: LCM = 2³ × 3² × 5 = 8 × 9 × 5 = 360. Therefore, the answer is 360.

Question 14

The HCF of two numbers is 8 and their LCM is 96. How many pairs of numbers satisfy this condition?

Accepted Answer

Step 1: If HCF = 8, both numbers can be written as 8a and 8b where HCF(a,b) = 1. Step 2: LCM(8a, 8b) = 8 × LCM(a,b) = 8ab = 96. Step 3: ab = 12. Step 4: Pairs (a,b) where HCF(a,b) = 1 and ab = 12: (1,12), (3,4). Step 5: Corresponding number pairs: (8,96) and (24,32). Answer: 2 pairs.

Question 15

Find the LCM of 24, 36, and 48.

Accepted Answer

Step 1: Prime factorize each number. 24 = 2³ × 3, 36 = 2² × 3², 48 = 2⁴ × 3. Step 2: LCM is the product of highest powers of all prime factors. Step 3: LCM = 2⁴ × 3² = 16 × 9 = 144. Therefore, the answer is 144.

Question 16

Two bells ring at intervals of 18 minutes and 24 minutes respectively. If they ring together at 9:00 AM, at what time will they ring together again?

Accepted Answer

Step 1: Find LCM(18, 24). 18 = 2 × 3², 24 = 2³ × 3. Step 2: LCM = 2³ × 3² = 8 × 9 = 72 minutes. Step 3: 9:00 AM + 72 minutes = 9:00 AM + 1 hour 12 minutes = 10:12 AM.

Question 17

The HCF of two numbers is 15. If the numbers are 45 and 75, verify the HCF is correct. What is their LCM?

Accepted Answer

Step 1: Verify HCF(45, 75): 45 = 3² × 5, 75 = 3 × 5². HCF = 3 × 5 = 15 ✓. Step 2: LCM = 3² × 5² = 9 × 25 = 225. Step 3: Check: HCF × LCM = 15 × 225 = 3375 = 45 × 75 ✓.

Question 18

The HCF of two numbers is 15. If the numbers are 45 and 75, verify which statement is correct.

Accepted Answer

Step 1: Find HCF of 45 and 75 using prime factorization. 45 = 3² × 5, 75 = 3 × 5². Step 2: HCF = 3 × 5 = 15. Step 3: The given HCF of 15 matches our calculation. Therefore, the statement is verified as correct, and the answer is 15.

Question 19

The LCM of two numbers is 168 and their HCF is 12. If the difference between the two numbers is 60, find the larger number.

Accepted Answer

Step 1: Let numbers be 12a and 12b where HCF(a,b) = 1. Step 2: LCM = 12ab = 168, so ab = 14. Step 3: Coprime pairs with product 14: (1,14) and (2,7). Step 4: For (1,14): numbers are 12 and 168, difference = 156 (not 60). Step 5: For (2,7): numbers are 24 and 84, difference = 60 ✓. Step 6: Larger number = 84. Therefore, the answer is 84.

Question 20

Three bells ring at intervals of 12, 18, and 24 minutes respectively. If they all ring together at 9:00 AM, at what time will they ring together again?

Accepted Answer

Step 1: Find LCM of 12, 18, and 24. Step 2: Prime factorization: 12 = 2² × 3, 18 = 2 × 3², 24 = 2³ × 3. Step 3: LCM = 2³ × 3² = 8 × 9 = 72 minutes. Step 4: 9:00 AM + 72 minutes = 10:12 AM.

Question 21

The HCF of two numbers is 8. If the numbers are in the ratio 3:5, what is the sum of the two numbers?

Accepted Answer

Step 1: Since HCF is 8 and numbers are in ratio 3:5, the numbers can be written as 3k and 5k where k is a positive integer. Step 2: HCF(3k, 5k) = k × HCF(3, 5) = k × 1 = k. Step 3: Given HCF = 8, so k = 8. Step 4: Numbers are 3(8) = 24 and 5(8) = 40. Step 5: Sum = 24 + 40 = 64.

Question 22

Three bells ring at intervals of 18 minutes, 24 minutes, and 30 minutes respectively. If they all ring together at 9:00 AM, at what time will they ring together again?

Accepted Answer

Step 1: Find LCM of 18, 24, and 30. Step 2: Prime factorization: 18 = 2 × 3², 24 = 2³ × 3, 30 = 2 × 3 × 5. Step 3: LCM = 2³ × 3² × 5 = 8 × 9 × 5 = 360 minutes. Step 4: Convert to hours: 360 ÷ 60 = 6 hours. Step 5: 9:00 AM + 6 hours = 3:00 PM. Therefore, the answer is 3:00 PM.

Question 23

The HCF of two numbers is 16 and the sum of the two numbers is 144. If one number is 48, what is the other number?

Accepted Answer

Step 1: Since HCF is 16, both numbers are multiples of 16. Let the numbers be 16a and 16b where HCF(a,b) = 1. Step 2: One number is 48, so 16a = 48, giving a = 3. Step 3: Sum of numbers: 16a + 16b = 144, so 48 + 16b = 144. Step 4: 16b = 96, so b = 6. Step 5: Other number = 16 × 6 = 96. Therefore, the answer is 96.

Question 24

Two numbers have an HCF of 18 and an LCM of 540. How many such pairs of numbers exist?

Accepted Answer

Step 1: If HCF = 18, then numbers are 18m and 18n where HCF(m,n) = 1. Step 2: LCM = 18 × m × n = 540, so m × n = 30. Step 3: Find coprime pairs (m,n) whose product is 30. Step 4: 30 = 2 × 3 × 5. Coprime factor pairs: (1,30), (2,15), (3,10), (5,6). Step 5: Count = 4 pairs. Therefore, the answer is 4.

Question 25

Three numbers are in the ratio 2:3:4. Their LCM is 120. What is the sum of the three numbers?

Accepted Answer

Step 1: Let the numbers be 2k, 3k, and 4k. Step 2: Find LCM(2k, 3k, 4k) = LCM(2,3,4) × k = 12k. Step 3: Given LCM = 120, so 12k = 120, giving k = 10. Step 4: Numbers are 2×10 = 20, 3×10 = 30, 4×10 = 40. Step 5: Sum = 20 + 30 + 40 = 90. Therefore, the answer is 90.

Question 26

Three bells ring at intervals of 18 minutes, 24 minutes, and 36 minutes respectively. If they all ring together at 9:00 AM, at what time will they ring together again?

Accepted Answer

Step 1: Find LCM of 18, 24, and 36. Prime factorizations: 18 = 2 × 3², 24 = 2³ × 3, 36 = 2² × 3². Step 2: LCM = 2³ × 3² = 8 × 9 = 72 minutes. Step 3: Add 72 minutes to 9:00 AM: 9:00 AM + 72 minutes = 10:12 AM. Therefore, they ring together again at 10:12 AM.

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