Least Common Multiple: What It Is, How to Calculate It, and Examples
Least Common Multiple (LCM): What It Is, How to Calculate It, and Examples
The least common multiple, commonly abbreviated as LCM, is the smallest positive integer that is a multiple of two or more numbers at the same time. It is an important mathematical tool for solving divisibility problems, organizing quantities that repeat at regular intervals, and working with fractions that have different denominators.
In this article, you will learn what the LCM is, how to find it by listing multiples, how to calculate it using prime factorization, and how to solve application problems step by step.
What Is the Least Common Multiple?
To understand the LCM, we first need to remember what a multiple is. The multiples of a number are obtained by multiplying that number by whole numbers.
A common multiple is a number that appears in the multiples of two or more numbers.
For example, consider the multiples of 4 and 6:
Multiples of 4:
4, 8, 12, 16, 20, 24, 28, 32, 36, ...
Multiples of 6:
6, 12, 18, 24, 30, 36, 42, ...
The common multiples are 12, 24, 36, ...
The smallest one is:
Therefore, the least common multiple is the smallest positive common multiple of the numbers being considered.
What Is the LCM Used For?
The least common multiple allows us to find a number that is exactly divisible by two or more quantities. For this reason, it appears in many mathematical situations.
- Solving problems involving events that repeat periodically.
- Finding a common denominator when adding or subtracting fractions.
- Working with divisibility problems.
- Comparing repeating cycles or intervals.
- Solving mathematical problems using prime factorization.
How to Calculate the Least Common Multiple
There are several methods for calculating the LCM. The two most commonly used methods are listing multiples and prime factorization. Another method is simultaneous division.
Method 1: List the Multiples
This method is especially useful when working with small numbers.
Multiples of 6:
6, 12, 18, 24, 30, 36, ...
Multiples of 8:
8, 16, 24, 32, 40, ...
The first number that appears in both lists is 24.
Check:
24 ÷ 6 = 4
24 ÷ 8 = 3
Both divisions are exact, so 24 is a multiple of both 6 and 8.
Method 2: Prime Factorization
When the numbers are larger, it is often more efficient to use prime factorization.
The procedure consists of expressing each number as a product of prime factors and then taking all the prime factors that appear, using the highest exponent for each factor.
1. Factor each number into primes.
2. Select the prime factors using their highest exponents.
For the number 2, the exponents are 2 and 1. We take 2².
For the number 3, the exponents are 1 and 2. We take 3².
3. Perform the calculation.
Check:
36 ÷ 12 = 3
36 ÷ 18 = 2
Method 3: Simultaneous Division
Another way to calculate the LCM is to divide the numbers simultaneously by prime factors until all the numbers become 1.
| 12 | 18 | Prime divisor |
|---|---|---|
| 6 | 9 | 2 |
| 3 | 9 | 2 |
| 1 | 3 | 3 |
| 1 | 1 | 3 |
Now multiply the prime divisors used:
Least Common Multiple of Three Numbers
The same procedure can be applied when we need to calculate the LCM of three or more numbers.
First, factor the three numbers into prime factors:
Now select each prime factor using its highest exponent:
Calculate:
Check:
72 ÷ 8 = 9
72 ÷ 12 = 6
72 ÷ 18 = 4
Therefore, 72 is a multiple of all three numbers.
How Can You Check Whether the LCM Is Correct?
Once you have obtained the result, you can check it by dividing the LCM by each of the original numbers.
If the result is exactly divisible by all the numbers considered, then you have found a common multiple.
In addition, for it to be the least common multiple, there must be no smaller positive common multiple.
LCM and GCD: What Is the Difference?
The LCM and the GCD are both related to divisibility, but they answer different mathematical questions.
| Concept | What are we looking for? |
|---|---|
| LCM | The smallest positive common multiple. |
| GCD | The greatest common divisor. |
The least common multiple is:
The greatest common divisor is:
The fundamental difference is that the LCM looks for a multiple, while the GCD looks for a divisor.
LCM Application Problems
The LCM is especially useful when several events repeat and we want to determine when they will occur together again.
Situation: One alarm rings every 6 minutes and another alarm rings every 8 minutes. If both alarms ring at the same time now, after how many minutes will they ring together again?
Using the LCM to Add Fractions
The least common multiple can also be used to find a common denominator when fractions have different denominators.
First, find the LCM of the denominators:
Convert both fractions to denominator 12:
Now add:
Common Mistakes When Calculating the LCM
The LCM looks for a common multiple, while the GCD looks for a common divisor.
The LCM must be the smallest positive common multiple, not simply any number that is divisible by the given numbers.
When calculating the LCM using prime factorization, each prime factor must be taken with its highest exponent.
It is a good practice to divide the LCM by each original number to verify that all divisions are exact.
Least Common Multiple Exercises
Practice the procedure with the following exercises:
Calculate the LCM of 4 and 10.
Calculate the LCM of 9 and 12.
Calculate the LCM of 10, 15, and 20.
Take the prime factors using their highest exponents:
Frequently Asked Questions About the LCM
What is the least common multiple?
It is the smallest positive common multiple of two or more numbers.
How is the least common multiple calculated?
It can be calculated by listing multiples, using prime factorization, or using simultaneous division.
How do you find the LCM of two numbers?
You can list their multiples until you find the first common one, or factor both numbers into primes and select each prime factor using its highest exponent.
How do you find the LCM of three numbers?
Factor all three numbers into prime factors and take each prime factor using the highest exponent that appears in any of the factorizations.
What is the difference between LCM and GCD?
The LCM finds the smallest positive common multiple, while the GCD finds the greatest common divisor.
What is the least common multiple used for?
It can be used to solve repeating-cycle problems, divisibility problems, and operations involving fractions with different denominators.
How do you solve least common multiple word problems?
First identify the quantities or events that repeat. Then determine the numbers involved and calculate their LCM. Finally, interpret the result according to the context of the problem.
Conclusion
The least common multiple is the smallest positive integer that is a multiple of two or more numbers at the same time. Understanding multiples, common multiples, and divisibility makes it easier to use the LCM correctly.
For small numbers, listing multiples can be a simple procedure. For larger numbers, prime factorization provides a systematic and efficient method: take every necessary prime factor using the highest exponent that appears.
The LCM has important applications in repeating-cycle problems and operations involving fractions. The key is to identify exactly what the problem is asking for and verify that the result is divisible by all the numbers involved.
Keep Learning with Frecuencia Educativa
This topic is part of the episode:
T2 E5 | Divisibility, Prime Numbers, LCM, and GCD | Frecuencia Educativa
In this episode, you will find an integrated explanation of four fundamental topics in Arithmetic, as well as the connections between them.
In this article, we take a deeper look at the specific topic of Least Common Multiple: What It Is, How to Calculate It, and Examples.
🎙️ Listen to the full episode here:
▶️ T2 E5 | Divisibility, Prime Numbers, LCM, and GCD
Module Infographic
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Academic Resource Profile
Collection
Mathematics
Knowledge Center
Arithmetic
Knowledge Area
Divisibility
Topic
Least Common Multiple (LCM): What It Is, How to Calculate It, and Examples
Level
Basic–Intermediate
Estimated Reading Time
10–12 minutes
Resource Type
Master Article
Skills Developed
Understand the concept of a multiple and a common multiple.
Identify the least common multiple of two or more numbers.
Calculate the LCM by listing multiples.
Calculate the LCM using prime factorization.
Calculate the LCM using the simultaneous division method.
Determine the LCM of three or more numbers.
Verify whether a result is the least common multiple.
Distinguish between the LCM and the GCD.
Apply the LCM to problems involving cycles or recurring events.
Use the LCM to find common denominators when performing operations with fractions.
Recognize common mistakes when calculating the LCM.
Relate the LCM to divisibility, prime numbers, and factorization.
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