Question 1

Find the HCF of 48 and 64.

Accepted Answer

Step 1: Find prime factorization of 48 = 2⁴ × 3. Step 2: Find prime factorization of 64 = 2⁶. Step 3: HCF is the product of lowest powers of common prime factors = 2⁴ = 16. Therefore, the answer is 16.

Question 2

The product of two numbers is 1800 and their HCF is 15. What is their LCM?

Accepted Answer

Step 1: Use the formula: HCF × LCM = Product of two numbers. Step 2: 15 × LCM = 1800. Step 3: LCM = 1800 ÷ 15 = 120. Therefore, the answer is 120.

Question 3

Find the HCF of 36, 54, and 72.

Accepted Answer

Step 1: Prime factorization: 36 = 2² × 3², 54 = 2 × 3³, 72 = 2³ × 3². Step 2: HCF is the product of lowest powers of common prime factors = 2¹ × 3² = 2 × 9 = 18. Therefore, the answer is 18.

Question 4

What is the LCM of 12, 18, and 24?

Accepted Answer

Step 1: Prime factorization: 12 = 2² × 3, 18 = 2 × 3², 24 = 2³ × 3. Step 2: LCM is the product of highest powers of all prime factors = 2³ × 3² = 8 × 9 = 72. Therefore, the answer is 72.

Question 5

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. 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 6

The HCF of two numbers is 15. If the numbers are in the ratio 3:5, find the larger number.

Accepted Answer

Step 1: Let the two numbers be 3x and 5x. Step 2: HCF of 3x and 5x is x (since HCF of 3 and 5 is 1). Step 3: Given HCF = 15, so x = 15. Step 4: Larger number = 5x = 5 × 15 = 75.

Question 7

A rectangular room has dimensions 48 m × 36 m. Square tiles of the largest possible size are to be laid without cutting. What is the side length of each tile, and how many tiles are needed?

Accepted Answer

Step 1: Find HCF(48, 36) for the largest tile size. Step 2: 48 = 2⁴ × 3, 36 = 2² × 3. Step 3: HCF = 2² × 3 = 12 m. Step 4: Number of tiles = (48 × 36) ÷ (12 × 12) = 1728 ÷ 144 = 12 tiles.

Question 8

Two numbers have an HCF of 15 and an LCM of 450. How many pairs of such numbers exist?

Accepted Answer

Step 1: If HCF = 15, both numbers can be written as 15m and 15n where HCF(m, n) = 1 (coprime). Step 2: LCM(15m, 15n) = 15 × LCM(m, n) = 15 × m × n (since m and n are coprime). Step 3: So 15 × m × n = 450, giving m × n = 30. Step 4: Find coprime pairs (m, n) whose product is 30. Prime factorization: 30 = 2 × 3 × 5. Step 5: Coprime pairs: (1, 30), (2, 15), (3, 10), (5, 6). That's 4 pairs. Step 6: Since order matters for 'pairs', we count (1,30) and (30,1) as one unordered pair. Answer: 4 pairs.

Question 9

Four bells ring at intervals of 6, 8, 12, and 18 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(6, 8, 12, 18). Prime factorizations: 6 = 2×3, 8 = 2³, 12 = 2²×3, 18 = 2×3². Step 2: LCM = 2³ × 3² = 8 × 9 = 72 minutes. Step 3: 72 minutes = 1 hour 12 minutes. Step 4: 9:00 AM + 1 hour 12 minutes = 10:12 AM.

Question 10

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

Accepted Answer

Step 1: Let the three numbers be 2k, 3k, and 4k where k is the HCF. Step 2: Find LCM(2k, 3k, 4k). Since LCM(2, 3, 4) = 12, we have LCM(2k, 3k, 4k) = 12k. Step 3: Given LCM = 144, so 12k = 144, thus k = 12. Therefore, HCF = 12.

SSC GD Constable 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