Maths-4 HCF and LCM : Methods and Step-by-Step Solutions for Competitive Exams SSC, BSSC, Bank, Railways)

Maths-4 HCF and LCM : Methods and Step-by-Step Solutions for Competitive Exams SSC, BSSC, Bank, Railways)

HCF and LCM from the basics. These are foundational concepts in arithmetic and number theory, and they appear often in competitive exams like CDS, SSC, and UPSC.

What is HCF and LCM?

HCF (Highest Common Factor)

• The largest number that divides two or more numbers exactly.
• Also called GCD (Greatest Common Divisor).

LCM (Least Common Multiple)

• The smallest number that is a multiple of two or more numbers.

Basic Method to Find HCF and LCM

1. Prime Factorization Method

Let’s take two numbers: 12 and 18

Step 1: Prime factorize both numbers

• 12 = 2 × 2 × 3
• 18 = 2 × 3 × 3

Step 2: HCF = Product of common prime factors

• Common: 2 × 3 = 6

Step 3: LCM = Product of all prime factors (take highest powers)

• LCM = 2² × 3² = 36

2. Division Method (for HCF)

Let’s take 36 and 60

Step 1: Divide larger by smaller: 60 ÷ 36 = remainder 24
Step 2: Divide 36 by 24 → remainder 12
Step 3: Divide 24 by 12 → remainder 0
✅ When remainder is 0, divisor is HCF = 12

3. Formula Method

If you know HCF and LCM of two numbers:

HCF × LCM = Product of the numbers

Example:
If HCF of 12 and 18 is 6, then
LCM = (12 × 18)/6 = 36

Examples

✅ Example 1: Find HCF and LCM of 8 and 12

• 8 = 2 × 2 × 2
• 12 = 2 × 2 × 3
• HCF = 2 × 2 = 4
• LCM = 2³ × 3 = 24

✅ Example 2: Find HCF and LCM of 15 and 25

• 15 = 3 × 5
• 25 = 5 × 5
• HCF = 5
• LCM = 3 × 5² = 75

✅ Example 3: Find HCF and LCM of 9, 12, and 15

• 9 = 3 × 3
• 12 = 2 × 2 × 3
• 15 = 3 × 5
• Common factor = 3 → HCF = 3
• LCM = 2² × 3² × 5 = 180

Shortcut Tips

Interactive HCF and LCM quiz with multiple-choice questions and step-by-step solutions. Try answering each question first, then check the solution below it.

HCF and LCM Quiz – 20 Questions with Solutions

🔹 HCF Section (Questions 1–10)

1. What is the HCF of 24 and 36?

1. 6 B) 12 C) 18 D) 24
   ✅ Correct Answer: B) 12
   Solution:
   24 = 2³ × 3  36 = 2² × 3²
   Common = 2² × 3 = 12

2. What is the HCF of 45 and 75?

1. 5 B) 15 C) 25 D) 30
   ✅ Correct Answer: B) 15
   Solution:
   45 = 3² × 5  75 = 3 × 5²
   Common = 3 × 5 = 15

3. Find the HCF of 18, 27, and 36

1. 3 B) 6 C) 9 D) 12
   ✅ Correct Answer: C) 9
   Solution:
   Common factor among all = 3² = 9

4. Two ropes are 12 m and 18 m long. What is the greatest length to cut both without leftover?

1. 3 B) 6 C) 9 D) 12
   ✅ Correct Answer: B) 6
   Solution:
   HCF of 12 and 18 = 6

5. A teacher has 36 pencils and 60 pens. What is the largest number of students who can get equal sets?

1. 6 B) 12 C) 18 D) 24
   ✅ Correct Answer: B) 12
   Solution:
   HCF of 36 and 60 = 12

6. What is the HCF of 48 and 64?

1. 8 B) 12 C) 16 D) 24
   ✅ Correct Answer: C) 16
   Solution:
   48 = 2⁴ × 3  64 = 2⁶
   Common = 2⁴ = 16

7. What is the HCF of 30, 45, and 60?

1. 5 B) 10 C) 15 D) 20
   ✅ Correct Answer: C) 15
   Solution:
   Common prime factors = 3 × 5 = 15

8. A gardener has 30 tulips and 45 roses. What is the maximum number of bouquets he can make?

1. 5 B) 10 C) 15 D) 20
   ✅ Correct Answer: C) 15
   Solution:
   HCF of 30 and 45 = 15

9. Find the HCF of 20, 25, and 30

1. 5 B) 10 C) 15 D) 20
   ✅ Correct Answer: A) 5
   Solution:
   Common factor = 5

10. A rectangular floor is 48 m by 60 m. What is the largest square tile that can be used without cutting?

1. 6 B) 12 C) 24 D) 48
   ✅ Correct Answer: B) 12
   Solution:
   HCF of 48 and 60 = 12

🔹 LCM Section (Questions 11–20)

11. What is the LCM of 12 and 18?

1. 24 B) 36 C) 48 D) 54
   ✅ Correct Answer: B) 36
   Solution:
   LCM = 2² × 3² = 36

12. What is the LCM of 6, 8, and 12?

1. 12 B) 24 C) 48 D) 96
   ✅ Correct Answer: B) 24
   Solution:
   LCM = 2³ × 3 = 24

13. Two bells ring every 4 and 6 minutes. When will they ring together next?

1. 10 B) 12 C) 14 D) 16
   ✅ Correct Answer: B) 12
   Solution:
   LCM of 4 and 6 = 12 minutes

14. A machine beeps every 8, 12, and 16 seconds. When will all beep together?

1. 24 B) 36 C) 48 D) 60
   ✅ Correct Answer: C) 48
   Solution:
   LCM = 2⁴ × 3 = 48 seconds

15. What is the LCM of 9, 12, and 15?

1. 90 B) 120 C) 180 D) 240
   ✅ Correct Answer: C) 180
   Solution:
   LCM = 2² × 3² × 5 = 180

16. A boy runs around a circular track in 12 minutes, another in 18 minutes. After how many minutes will they meet again?

1. 24 B) 30 C) 36 D) 48
   ✅ Correct Answer: C) 36
   Solution:
   LCM of 12 and 18 = 36 minutes

17. What is the LCM of 10, 20, and 25?

1. 50 B) 75 C) 100 D) 125
   ✅ Correct Answer: C) 100
   Solution:
   LCM = 2² × 5² = 100

18. Three traffic lights blink every 15, 20, and 30 seconds. When will they blink together?

1. 45 B) 60 C) 75 D) 90
   ✅ Correct Answer: B) 60
   Solution:
   LCM = 2² × 3 × 5 = 60 seconds

19. A gardener waters plants every 5, 10, and 20 days. After how many days will he water all on the same day again?

1. 10 B) 15 C) 20 D) 25
   ✅ Correct Answer: C) 20
   Solution:
   LCM = 2² × 5 = 20 days

20. If HCF of two numbers is 6 and their product is 216, what is their LCM?

1. 36 B) 54 C) 72 D) 90
   ✅ Correct Answer: C) 36
   Solution:
   LCM = Product ÷ HCF = 216 ÷ 6 = 36