Question 1

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 factorize: 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.

Question 2

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 = highest power of each prime = 2⁴ × 3² = 16 × 9 = 144.

Question 3

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

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 is correct. Step 4: Both numbers are divisible by 15: 45 ÷ 15 = 3, 75 ÷ 15 = 5.

Question 4

Three ropes of lengths 48 m, 60 m, and 72 m are to be cut into equal pieces without any waste. What is the maximum length of each piece?

Accepted Answer

Step 1: Find HCF of 48, 60, and 72. Prime factorize: 48 = 2⁴ × 3, 60 = 2² × 3 × 5, 72 = 2³ × 3². Step 2: HCF = 2² × 3 = 4 × 3 = 12 m. Step 3: Each piece will be 12 m long.

Question 5

The HCF of three numbers is 6. If the numbers are in the ratio 2:3:4, and their LCM is 360, what is the largest number?

Accepted Answer

Step 1: Since HCF is 6, numbers are 6a, 6b, 6c where HCF(a,b,c) = 1. Step 2: Ratio 2:3:4 means a=2, b=3, c=4. Step 3: Numbers are 12, 18, 24. Step 4: Verify LCM(12, 18, 24): Prime factors: 12=2²×3, 18=2×3², 24=2³×3. LCM = 2³×3² = 72. Step 5: But given LCM is 360, so numbers must be scaled. Step 6: 360 ÷ 72 = 5, so multiply each by 5: Numbers are 60, 90, 120. Step 7: Largest = 120.

Question 6

The HCF of two numbers is 15. If the numbers are in the ratio 3:5, what is the sum of the two numbers?

Accepted Answer

Step 1: Since HCF is 15, the numbers can be written as 15a and 15b where HCF(a,b) = 1. Step 2: Given ratio is 3:5, so a = 3 and b = 5. Step 3: The two numbers are 15 × 3 = 45 and 15 × 5 = 75. Step 4: Sum = 45 + 75 = 120.

Question 7

Two numbers have an HCF of 8 and an LCM of 240. How many pairs of such numbers are possible?

Accepted Answer

Step 1: If HCF = 8, both numbers are multiples of 8. Let numbers be 8m and 8n where HCF(m,n) = 1. Step 2: LCM(8m, 8n) = 8 × LCM(m,n) = 240, so LCM(m,n) = 30. Step 3: Find coprime pairs (m,n) whose LCM is 30. Step 4: 30 = 2 × 3 × 5. Coprime pairs: (1,30), (2,15), (3,10), (5,6). Step 5: Total pairs = 4.

Question 8

Three bells ring at intervals of 18, 24, 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. Step 2: Prime factorization: 18 = 2 × 3², 24 = 2³ × 3, 36 = 2² × 3². Step 3: LCM = 2³ × 3² = 8 × 9 = 72 minutes. Step 4: 9:00 AM + 72 minutes = 10:12 AM.

Question 9

The HCF of two numbers is 16. Their LCM is 240. 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. Step 2: LCM(16m, 16n) = 16 × LCM(m,n) = 16 × m × n = 240. Step 3: m × n = 15. Step 4: Find coprime pairs (m,n) whose product is 15: (1,15) and (3,5). Step 5: Numbers are (16,240) and (48,80). Therefore, 2 pairs satisfy the condition.

Question 10

Two numbers have an HCF of 14 and an LCM of 420. If the sum of the two numbers is 154, find the difference between them.

Accepted Answer

Step 1: Let the numbers be 14m and 14n where HCF(m,n) = 1. Step 2: LCM = 14mn = 420, so mn = 30. Step 3: Sum = 14m + 14n = 154, so m + n = 11. Step 4: Solve: m and n are roots of t² - 11t + 30 = 0. Step 5: (t-5)(t-6) = 0, so m = 5, n = 6. Step 6: Numbers are 70 and 84. Step 7: Difference = 84 - 70 = 14.

Question 11

Three numbers are in the ratio 2:3:4. Their LCM is 144. Find the HCF of the three numbers.

Accepted Answer

Step 1: Let the numbers be 2k, 3k, and 4k. Step 2: Find LCM(2k, 3k, 4k). Step 3: LCM = k × LCM(2, 3, 4) = k × 12. Step 4: k × 12 = 144, so k = 12. Step 5: Numbers are 24, 36, 48. Step 6: HCF(24, 36, 48) = 12. Therefore, the HCF is 12.

IBPS RRB PO LCM and HCF

LCM and HCF— Rules & Concept

Key Points to Remember

Exam-Specific Tips

LCM and HCF — Practice Questions

60-Second Revision — LCM and HCF