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LCM Questions

LCM questions are a crucial part of elementary and advanced mathematics. Whether you're preparing for school exams or competitive tests, understanding what is LCM in math and learning how to solve LCM word problems can give you a significant edge.

The least common multiple (LCM) helps us find the smallest multiple that is common to two or more numbers. It plays a vital role in topics like fractions, time intervals, and number systems. This guide will explain the LCM method, offer LCM examples, and present a variety of LCM questions with solutions to help you master the concept.

Table of Contents

What is LCM in Math?
Least Common Multiple Formula
LCM Questions With Solutions
LCM Word Problems
LCM Practice Problems
Real-Life Applications of LCM
Tips to Solve LCM Questions Faster
Conclusion
FAQs on LCM questions

What is LCM in Math?

In mathematics, LCM stands for Least Common Multiple. It is the smallest number that is a multiple of two or more given numbers.

For example:

The multiples of 4 are: 4, 8, 12, 16, 20, 24...
The multiples of 6 are: 6, 12, 18, 24, 30...

The least common multiple of 4 and 6 is 12, because it is the smallest number that appears in both lists.

Understanding what is LCM in math lays the foundation for solving a variety of LCM questions effectively.

Least Common Multiple Formula

Formula / Concept	Expression / Rule	Example / Use Case
1. Basic LCM Formula	LCM(a, b) = (a × b) / HCF(a, b)	LCM(252, 594) = (252 × 594) / 18 = 8316
2. Relationship between LCM and HCF	LCM × HCF = Product of two numbers	252 × 594 = LCM × HCF ⇒ 8316 × 18
3. LCM when same remainder R	Number = LCM(n₁, n₂, n₃, ...) + R	Number = LCM(35, 56, 91) + 7 = 3647
4. LCM when different remainders R₁, R₂...	Number = LCM(n₁, n₂, ...) − R, where R = n₁ − R₁ = n₂ − R₂ = ...	28–8 = 20 and 32–12 = 20 ⇒ Number = LCM(28, 32) − 20 = 204
5. LCM of polynomials	Multiply all irreducible factors raised to highest powers	192(x – 1)(x + 1)²(x + 2)²(x + 3)³
6. LCM of fractions	LCM = LCM of numerators / HCF of denominators	LCM(⅔, ¾, 7/2) = LCM(2, 3, 7) / HCF(3, 4, 2) = 42
7. LCM of decimals	Convert to integers, find LCM, place decimal back according to original places	LCM(22.5, 3.5, 0.55) = 1732.5
8. LCM of algebraic expressions	Multiply all variables and constants raised to the highest powers	Example: x³y²z × x²y³ = x³y³z
9. Minimum number divisible by all	Least number = LCM of all given numbers	LCM(56, 58) = 1624 books needed
10. Repetition of event interval	Time interval = LCM of time gaps	LCM(72, 50) = 1800 seconds = 30 minutes
11. Largest 4-digit number with specific remainder	x = [Largest 4-digit number] − remainder adjustment	9999 – 171 + 3 = 9831

This least common multiple formula works especially well for smaller numbers and saves time during LCM questions in exams.

LCM Questions With Solutions

Let’s explore some real LCM questions with solutions to build mastery:

Listing Multiples Method

This is a basic method where you list the multiples of each number and find the least common multiple.

Example 1:

Find the LCM of 3 and 5 using the listing method.

Step 1: List the multiples of 3:
3, 6, 9, 12, 15, 18, 21, 24...

Step 2: List the multiples of 5:
5, 10, 15, 20, 25...

Step 3: Find the common multiples:
Common multiples = 15, 30...

Step 4: Choose the least one:
LCM = 15

Example 2:

Find the LCM of 4 and 6.

Step 1: Multiples of 4:
4, 8, 12, 16, 20, 24...

Step 2: Multiples of 6:
6, 12, 18, 24...

Step 3: Common multiples:
12, 24...

Step 4: Least common multiple:
LCM = 12

Example 3:

Find the LCM of 7 and 8.

Step 1: Multiples of 7:
7, 14, 21, 28, 35, 42, 49, 56

Step 2: Multiples of 8:
8, 16, 24, 32, 40, 48, 56

Step 3: Common multiple:
56

Step 4:
LCM = 56

Example 4:

Find the LCM of 6 and 9.

Step 1: Multiples of 6:
6, 12, 18, 24, 30, 36

Step 2: Multiples of 9:
9, 18, 27, 36

Step 3: Common multiples:
18, 36...

Step 4:
LCM = 18

Prime Factorization Method

This method uses prime factor trees and selects the highest power of each prime.

Example 1:

Find the LCM of 12 and 15.

Step 1: Prime factorization:
12 = 2² × 3
15 = 3 × 5

Step 2: Take the highest power of each prime:
2², 3, 5

Step 3: Multiply them:
LCM = 2² × 3 × 5 = 4 × 3 × 5 = 60

Example 2:

Find the LCM of 18 and 24.

Step 1: Prime factorization:
18 = 2 × 3²
24 = 2³ × 3

Step 2: Highest powers:
2³, 3²

Step 3: Multiply:
LCM = 2³ × 3² = 8 × 9 = 72

Example 3:

Find the LCM of 16 and 20.

Step 1: Prime factorization:
16 = 2⁴
20 = 2² × 5

Step 2: Highest powers:
2⁴, 5

Step 3: LCM = 2⁴ × 5 = 16 × 5 = 80

Example 4:

Find the LCM of 21 and 6.

Step 1: Prime factorization:
21 = 3 × 7
6 = 2 × 3

Step 2: Take the highest powers:
2, 3, 7

Step 3: Multiply:
LCM = 2 × 3 × 7 = 42

Division Method

Example 1: Find the LCM of 12, 15, and 20

Step 1: Write the numbers:
12 15 20

Step 2: Start dividing using prime numbers:

Prime	Numbers
2	6 15 10
2	3 15 5
3	1 5 5
5	1 1 1

LCM = 2 × 2 × 3 × 5 = 60

Example 2: Find the LCM of 8, 12, and 16

Start with the numbers:
8 12 16

Use division:

Prime	Numbers
2	4 6 8
2	2 3 4
2	1 3 2
2	1 3 1
3	1 1 1

LCM = 2 × 2 × 2 × 2 × 3 = 48

Example 3: Find the LCM of 10, 15, and 25

Start with:
10 15 25

Use division:

Prime	Numbers
2	5 15 25
3	5 15 25
5	1 3 5
3	1 1 5
5	1 1 1

LCM = 2 × 3 × 5 × 3 × 5 = 450

(Note: You can skip a prime if it doesn't divide any number; continue with the next.)

Example 4: Find the LCM of 6, 8, and 9

Start with:
6 8 9

Divide step by step:

Prime	Numbers
2	3 4 9
2	3 2 9
2	3 1 9
3	1 1 3
3	1 1 1

LCM = 2 × 2 × 2 × 3 × 3 = 72

LCM Word Problems

Understanding LCM word problems helps apply math in real-life situations. Let’s solve some:

Problem 1:

Three buses leave a station at intervals of 20, 30, and 40 minutes. After how much time will they all leave the station together again?

Find the LCM of 20, 30, and 40

Prime factorizations:

20 = 2² × 5
30 = 2 × 3 × 5
40 = 2³ × 5

LCM = 2³ × 3 × 5 = 120 minutes = 2 hours

Problem 2:

A gardener plants roses every 6 days and lilies every 8 days. In how many days will both be planted on the same day?

Find LCM of 6 and 8

6 = 2 × 3
8 = 2³
LCM = 2³ × 3 = 24 days

These LCM word problems train your mind for practical applications.

Problem 3:

A teacher assigns projects every 10 days, grades homework every 5 days, and administers tests every 7 days. How many days will it take her to complete all three tasks on the same day if she completed them all on Monday?

Solution:
Find the LCM of 5, 7, and 10
5 = 5
7 = 7
10 = 2 × 5
LCM = 2 × 5 × 7 = 70

Answer: After 70 days

Problem 4:

Three wheels rotate 12, 15, and 18 times per minute. After how many seconds will they align again at the starting point?

Solution:
Find the LCM of 12, 15, 18
12 = 2² × 3
15 = 3 × 5
18 = 2 × 3²
LCM = 2² × 3² × 5 = 180

LCM in minutes = 180 rotations
Since rotations are per minute, align after 1 minute (60 sec × 3 = 180 sec)
Answer: After 60 seconds

Problem 5:

Three lights flash every 8, 12, and 16 seconds respectively. They flash together now. After how many seconds will they flash together again?

Solution:
Find LCM of 8, 12, and 16
8 = 2³
12 = 2² × 3
16 = 2⁴
LCM = 2⁴ × 3 = 48

Answer: After 48 seconds

LCM Practice Problems

Try solving these LCM practice problems:

Find the LCM of 8 and 12.
What is the LCM of 15 and 25?
Find the least common multiple of 14, 28, and 35.
Two traffic lights blink every 48 and 60 seconds, respectively. When will they blink together again?
A number leaves a remainder of 5 when divided by 12 and 18. Find the smallest such number.
Find the LCM of 45, 60, and 75.
The bells toll every 30, 40, and 45 seconds. How often will all three toll together?
Find the LCM of 22.5, 3.5, and 0.55.
A boy exercises every 6 days, another every 9 days. If they start on the same day, when will they next exercise together?
A box of chocolates is to be packed such that each pack has the same number of chocolates, and the number should be divisible by 18, 24, and 30. Find the least such number.
Find the LCM of the polynomials:
a) (x + 2)(x + 3), (x + 2)(x + 4)
b) x² – 1 and x² – 4

Practicing such LCM questions regularly improves speed and accuracy.

Real-Life Applications of LCM

LCM questions are not just limited to exams; they help in many daily scenarios:

Traffic light synchronisation
Setting up activities or events
Resolving issues on time and at work
Controlling the cycles of inventory
Creating schedules for schools

These examples show that knowing what is LCM in math can even solve scheduling conflicts in real life!

Tips to Solve LCM Questions Faster

Know your primes: Learn all 50 prime numbers by heart.
Use shortcuts: When dealing with large numbers, use the least popular multiple formula.
Practice often: Every day, solve LCM practice problems.
Understand every technique: Recognise when to divide, factorise, or list.
Use the HCF-LCM relation: For two numbers,
LCM×HCF=Product of the two numbers

These tips help you handle all LCM questions with ease.

Conclusion

Mastering LCM questions is essential for success in math exams and real-life situations. From understanding what is LCM in math to applying the least common multiple formula and solving tricky LCM word problems, this guide has covered all aspects thoroughly.

Related Links

HCF and LCM Questions - Practice a variety of HCF and LCM questions to boost your understanding through hands-on problem solving.

Prime Factorization of HCF and LCM - Discover how to calculate HCF and LCM using prime factorization with step-by-step methods.

HCF and LCM - Grasp the core concepts of HCF and LCM with clear explanations and real-life examples.

HCF - Learn multiple techniques to find the Highest Common Factor easily and accurately.

Frequently Asked Questions on LCM questions

1. How to find LCM fast?

To find LCM quickly, use the formula:
LCM(a, b) = (a × b) / HCF(a, b)
Alternatively, use the prime factorization method for smaller numbers or the division method for 3 or more numbers.