What is the most common type of LCM question in exams?

The most common LCM questions are: (1) Product formula questions (HCF × LCM = Product of numbers), (2) Remainder-based questions (find the least number which when divided leaves a specific remainder), and (3) Word problems involving bells, buses, or repeating events.

How do I find the LCM of two numbers if their HCF and product are given?

Use the formula: LCM = (Product of numbers) ÷ HCF. For example, if product = 1530 and HCF = 15, then LCM = 1530 ÷ 15 = 102.

What is the trick for LCM questions with remainders?

If a number leaves the SAME remainder 'r' when divided by a, b, c, then the number is of the form LCM(a, b, c) + r. If the differences between divisors and remainders are the same, use LCM(a, b, c) - difference.

How do I identify LCM questions in competitive exams?

Look for keywords like 'together', 'at the same time', 'minimum', 'least', 'smallest', 'repeat', or 'coincide'. These indicate LCM problems. If the question asks for 'maximum' or 'greatest', it's an HCF problem.

What is the LCM of two co-prime numbers?

The LCM of two co-prime numbers is their product. For example, LCM(4, 5) = 20 because GCD(4, 5) = 1.

Top 10 LCM Questions Asked in Exams (with Solutions)

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Exam High-Yield Top 10 LCM Questions Asked in Exams (with Solutions)

LCM (Least Common Multiple) is one of the highest-weightage topics tested across Class 6-8 school exams and competitive entrance tests (SSC, Railways, Banking, Olympiads, and NTSE). This definitive guide breaks down the top 10 recurring question patterns with complete step-by-step solutions, visual diagrams, and mental math shortcuts.

1. Product Formula Questions (Diagram 1) 2. Remainder Questions (Diagram 2) 3. Simultaneous Word Problems 4. Co-Prime & Ratio Questions 5. Digit Boundary Questions 6. Advanced & Perfect Square Questions Common Traps & Mistakes Frequently Asked Questions Exam Glossary

1. Product Formula Questions (Most Common Pattern)

These questions test your understanding of the universal relationship between HCF, LCM, and two numbers: HCF × LCM = Product of the Two Numbers.

Diagram 1: The HCF-LCM Product Formula Triangle

For any two numbers A and B, the product of their HCF and LCM always equals A × B. Use this visual formula triangle to isolate any single unknown variable instantly.

Question 1: Finding LCM from HCF and Product

Question: The product of two numbers is 1,530 and their HCF is 15. Find the LCM of these numbers.

Formula: HCF × LCM = Product of two numbers
15 × LCM = 1530
LCM = 1530 ÷ 15 = 102
Answer: 102

Question 2: Finding One Unknown Number

Question: The HCF of two numbers is 81 and their LCM is 10,206. If one of the numbers is 729, find the other number.

Let the second number be x.
729 × x = 81 × 10,206
729 × x = 826,686
x = 826,686 ÷ 729 = 1,134
Answer: 1,134

Question 3: Finding the Other Number with Large LCM

Question: The LCM of two numbers is 1,680 and their HCF is 24. If one number is 240, find the other number.

Let the other number be y.
240 × y = 24 × 1,680
240 × y = 40,320
y = 40,320 ÷ 240 = 168
Answer: 168

2. Remainder-Based Questions (High Weightage)

Remainder questions are famous in competitive exams. There are two distinct cases:

Diagram 2: Remainder Problem Dual-Rule Model (+r vs -d)

Case 1: When remainder 'r' is identical for all divisors, add r to LCM. Case 2: When differences between each divisor and its remainder are constant 'd', subtract d from LCM.

Question 4: Same Remainder in Each Case

Question: Find the smallest number which when divided by 15, 20, and 25 leaves a remainder of 7 in each case.

Step 1: Find LCM(15, 20, 25)
15 = 3 × 5; 20 = 2² × 5; 25 = 5²
LCM = 2² × 3 × 5² = 4 × 3 × 25 = 300
Step 2: Add remainder 7
Required number = 300 + 7 = 307
Answer: 307

Question 5: Different Remainders with Constant Difference

Question: Find the least number which when divided by 4, 5, and 6 leaves remainders 1, 2, and 3 respectively.

Step 1: Check difference between each divisor and remainder:
4 - 1 = 3
5 - 2 = 3
6 - 3 = 3 (Common difference d = 3)
Step 2: Find LCM(4, 5, 6) = 2² × 3 × 5 = 60
Step 3: Subtract difference d:
Required number = 60 - 3 = 57
(Check: 57 ÷ 4 = 14 rem 1; 57 ÷ 5 = 11 rem 2; 57 ÷ 6 = 9 rem 3 ✓)
Answer: 57

3. Simultaneous Word Problems (Bells & Traffic Lights)

Classic exam questions centered on simultaneous repetition cycles.

Question 6: Bells Ringing Together

Question: Three bells ring at intervals of 6, 8, and 12 minutes respectively. If they toll together at 8:00 AM, at what time will they toll together next?

LCM(6, 8, 12) = 2³ × 3 = 24 minutes.
Next time = 8:00 AM + 24 minutes = 8:24 AM.
Answer: 8:24 AM

Question 7: Traffic Signals at Three Crossings

Question: Traffic lights at three crossings change after every 48, 72, and 108 seconds. If they change simultaneously at 7:00:00 AM, at what time will they change together again?

LCM(48, 72, 108) = 2⁴ × 3³ = 16 × 27 = 432 seconds.
432 seconds = 7 minutes and 12 seconds.
7:00:00 AM + 7 min 12 s = 7:07:12 AM.
Answer: 7:07:12 AM

Question 8: Buses Departing Depot Routes

Question: Three buses leave a depot at intervals of 15, 20, and 30 minutes. If they leave together at 6:00 AM, when will they next leave together?

LCM(15, 20, 30) = 2² × 3 × 5 = 60 minutes = 1 hour.
6:00 AM + 1 hour = 7:00 AM.
Answer: 7:00 AM

4. Co-Prime Numbers & Ratio Questions

Examiners frequently pair ratios with HCF to test multi-step reasoning.

Question 9: LCM of Co-Prime Numbers

Question: If two numbers a and b are co-prime, what is their LCM?

Because co-prime numbers share no common factors other than 1, their HCF = 1.
Formula: HCF × LCM = a × b
1 × LCM = a × b → LCM = a × b.
Answer: Product of the two numbers (a × b)

Question 10: Numbers in a Ratio with Known HCF

Question: Two numbers are in the ratio 4:5. If their HCF is 6, find their LCM.

Let the numbers be 4x and 5x.
Since HCF is the common factor, x = 6.
The numbers are 4 × 6 = 24 and 5 × 6 = 30.
LCM(24, 30) = 2³ × 3 × 5 = 8 × 3 × 5 = 120.
Shortcut: LCM = Ratio 1 × Ratio 2 × HCF = 4 × 5 × 6 = 120.
Answer: 120

5. Digit Boundary Questions (3-Digit & 4-Digit Limits)

These questions test applying LCM to find bounds in our decimal system.

Question 11: Smallest 3-Digit Number Divisible by 6, 8, and 12

Question: Determine the smallest 3-digit number exactly divisible by 6, 8, and 12.

LCM(6, 8, 12) = 24.
Smallest 3-digit number is 100.
100 ÷ 24 = 4 with remainder 4.
Smallest multiple ≥ 100 is 24 × 5 = 120.
Answer: 120

Question 12: Greatest 3-Digit Number Divisible by 8, 10, and 12

Question: Determine the greatest 3-digit number exactly divisible by 8, 10, and 12.

LCM(8, 10, 12) = 120.
Largest 3-digit number is 999.
999 ÷ 120 = 8 with remainder 39.
Subtract remainder: 999 - 39 = 960.
Answer: 960

6. Advanced & Tricky Exam Questions

Master these to guarantee an edge in competitive exams and math olympiads.

Question 13: Smallest Perfect Square Divisible by 10, 16, and 24

Question: Find the smallest square number that is divisible by 10, 16, and 24.

Step 1: Find LCM(10, 16, 24)
10 = 2 × 5; 16 = 2⁴; 24 = 2³ × 3
LCM = 2⁴ × 3¹ × 5¹ = 240.
Step 2: Inspect prime powers for odd exponents:
240 = 2⁴ × 3¹ × 5¹.
To make a perfect square, all prime exponents must be even.
Missing factors: 3¹ and 5¹ (i.e. multiply by 15).
Smallest square = 240 × 15 = 3,600 (since 60² = 3,600).
Answer: 3,600

Common Mistakes in LCM Exam Questions

Avoid these 5 frequent pitfalls to secure top marks:

❌ Exam Pitfall	Why It Is False	✅ Correct Technique
Applying product formula to 3 numbers	LCM(a, b, c) × HCF(a, b, c) does NOT equal a × b × c	Use prime factor powers or division ladder for 3+ numbers
Adding difference instead of subtracting	When divisors minus remainders is constant d, student adds d	Same remainder → LCM + r; Same difference → LCM - d
Multiplying ratio numbers by LCM directly	Confusing HCF multiplier with LCM multiplier	LCM of ratio A:B with HCF h is A × B × h
Neglecting unit conversions in time problems	Adding 432 seconds directly as 432 minutes	Divide by 60 to convert seconds to minutes & seconds first
Guessing perfect square by squaring the LCM	Squaring 240 gives 57,600, which is not the smallest square	Multiply by only the unpaired odd prime factors (240 × 15 = 3600)

Frequently Asked Questions

Answers to recurring student questions on LCM exam preparation.

What is the most frequently asked LCM question in exams?

The product formula question (HCF × LCM = Product) and the simultaneous bell ringing question appear in nearly 80% of school and competitive exams.

How do I find LCM when given ratio and HCF?

If two numbers are in ratio a:b and their HCF is h, their LCM is simply a × b × h.

What is the trick for remainder LCM questions?

If remainder r is constant: LCM + r. If difference d = (divisor - remainder) is constant: LCM - d.

Mathematical Glossary for Exams

High-yield terms for Class 6-8 and competitive exams.

Least Common Multiple (LCM): The smallest positive integer that is evenly divisible by all numbers in a given set.
Product Formula: The theorem stating that HCF(a, b) × LCM(a, b) = a × b, strictly valid for exactly two numbers.
Co-prime Pair: Two integers whose highest common factor is 1; their LCM always equals their product.
Constant Difference Property: When dividing by various divisors yields a uniform difference (divisor - remainder = d), the answer equals LCM - d.