Maths-4 LCM (Least Common Multiple) : Questions and Step-by-Step Solutions for Competitive Exams SSC, BSSC, Bank,Railways)
LCM (Least Common Multiple) from the very basics in easy, step-by-step style. This is super useful for solving problems involving time, repetition, or synchronization.
What is LCM (Least Commom Multiple )
LCM of two or more numbers is the smallest number that is a multiple of all the given numbers.
Step-by-Step Method to Find LCM
Method 1: Prime Factorization (Best for small numbers)
Example: Find LCM of 12 and 18
Step 1: Write prime factors
- 12 = 2 × 2 × 3
- 18 = 2 × 3 × 3
Step 2: Take all prime factors, using the highest power
- LCM = 2² × 3² = 36
Method 2: Listing Multiples (Good for small numbers)
Example: Find LCM of 4 and 6
Step 1: List multiples
- 4 → 4, 8, 12, 16, 20, 24…
- 6 → 6, 12, 18, 24…
Step 2: Find the first common multiple
- LCM = 12
Method 3: Division Method (Best for 3+ numbers)
Example: Find LCM of 12, 15, and 20
Step 1: Divide by common prime numbers step-by-step:
12 | 2
15 | 2
20 | 2
————
6 | 3
15 | 3
10 | 3
————
2 | 5
5 | 5
10 | 5
————
1 | 1
1 | 1
2 | 2
Step 2: Multiply all divisors:
LCM = 2 × 3 × 5 × 2 = 60
Quick Tips
- LCM is always equal to or greater than the largest number.
- Use LCM when things repeat or need synchronization (bells ringing, pipes filling, etc.).
- For two numbers:
LCM = (Product of numbers) ÷ HCF
LCM Practice Questions with Step-by-Step Solutions
🔹 Basic Level
1. Find the LCM of 4 and 6
Step 1: Prime factors
- 4 = 2 × 2
- 6 = 2 × 3
Step 2: Take all prime factors with highest powers
- LCM = 2² × 3 = 12
2. Find the LCM of 5 and 10
Step 1:
- 5 = 5
- 10 = 2 × 5
Step 2:
- LCM = 2 × 5 = 10
3. Find the LCM of 3 and 7
Step 1:
- 3 and 7 are prime numbers
Step 2:
- LCM = 3 × 7 = 21
4. Find the LCM of 8 and 12
Step 1:
- 8 = 2³
- 12 = 2² × 3
Step 2:
- LCM = 2³ × 3 = 24
5. Find the LCM of 9 and 15
Step 1:
- 9 = 3²
- 15 = 3 × 5
Step 2:
- LCM = 3² × 5 = 45
🔹 Intermediate Level
6. Find the LCM of 12 and 18
Step 1:
- 12 = 2² × 3
- 18 = 2 × 3²
Step 2:
- LCM = 2² × 3² = 36
7. Find the LCM of 16 and 20
Step 1:
- 16 = 2⁴
- 20 = 2² × 5
Step 2:
- LCM = 2⁴ × 5 = 80
8. Find the LCM of 14 and 35
Step 1:
- 14 = 2 × 7
- 35 = 5 × 7
Step 2:
- LCM = 2 × 5 × 7 = 70
9. Find the LCM of 21 and 28
Step 1:
- 21 = 3 × 7
- 28 = 2 × 2 × 7
Step 2:
- LCM = 2² × 3 × 7 = 84
10. Find the LCM of 24 and 36
Step 1:
- 24 = 2³ × 3
- 36 = 2² × 3²
Step 2:
- LCM = 2³ × 3² = 72
🔹 Three Numbers
11. Find the LCM of 3, 4, and 6
Step 1:
- 3 = 3
- 4 = 2²
- 6 = 2 × 3
Step 2:
- LCM = 2² × 3 = 12
12. Find the LCM of 5, 10, and 15
Step 1:
- 5 = 5
- 10 = 2 × 5
- 15 = 3 × 5
Step 2:
- LCM = 2 × 3 × 5 = 30
13. Find the LCM of 6, 8, and 12
Step 1:
- 6 = 2 × 3
- 8 = 2³
- 12 = 2² × 3
Step 2:
- LCM = 2³ × 3 = 24
14. Find the LCM of 9, 12, and 15
Step 1:
- 9 = 3²
- 12 = 2² × 3
- 15 = 3 × 5
Step 2:
- LCM = 2² × 3² × 5 = 180
15. Find the LCM of 10, 20, and 25
Step 1:
- 10 = 2 × 5
- 20 = 2² × 5
- 25 = 5²
Step 2:
- LCM = 2² × 5² = 100
🔹 Word Problems
16. Two bells ring every 4 and 6 minutes. When will they ring together next?
Step 1:
- LCM of 4 and 6 = 12 minutes
17. Three traffic lights blink every 15, 20, and 30 seconds. When will they blink together?
Step 1:
- LCM of 15, 20, 30 =
- 15 = 3 × 5
- 20 = 2² × 5
- 30 = 2 × 3 × 5
- LCM = 2² × 3 × 5 = 60 seconds
18. A boy runs around a circular track in 12 minutes, another in 18 minutes. After how many minutes will they meet again at the starting point?
Step 1:
- LCM of 12 and 18 =
- 12 = 2² × 3
- 18 = 2 × 3²
- LCM = 2² × 3² = 36 minutes
19. A machine beeps every 8, 12, and 16 seconds. When will all beep together?
Step 1:
- LCM of 8, 12, 16 =
- 8 = 2³
- 12 = 2² × 3
- 16 = 2⁴
- LCM = 2⁴ × 3 = 48 seconds
20. A gardener waters plants every 5, 10, and 20 days. After how many days will he water all on the same day again?
Step 1:
- LCM of 5, 10, 20 =
- 5 = 5
- 10 = 2 × 5
- 20 = 2² × 5
- LCM = 2² × 5 = 20 days
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