LCM calculator
LCM calculator computes The Least Common Multiple of two or more numbers using five methods. (division method, listing multiples method, prime factors, ladder method and the LCM formula). The calculator shows a full step-by-step solution of each method, along with a Venn diagram.
Find Least Common Multiple of 12 and 8 using prime factorisation method.
solution
LCM( 12, 8 ) = 24
explanation
Step 1: Write down factorisation of each number:
12 = 2 · 2 · 3
8 = 2 · 2 · 2
Step 2 : Match primes vertically:
| 12 | = | 2 | · | 2 | · | 3 | ||
| 8 | = | 2 | · | 2 | · | 2 |
Step 3 : Bring down numbers in each column and multiply to get LCM:
| 12 | = | 2 | · | 2 | · | 3 | ||||
| 8 | = | 2 | · | 2 | · | 2 | ||||
| LCM | = | 2 | · | 2 | · | 2 | · | 3 | = | 24 |
This solution can be visualized using a Venn diagram.
The LCM is equal to the product of all the numbers on the diagram.
Tutorial
What is the Least Common Multiple?
The least common multiple (LCM) is the smallest number that can be divided by all the given numbers. For example, LCM(8, 6) = 24, because 24 is divisible by 8 and 6, and it is the smallest such number.
How to calculate LCM
There are five common methods for calculating LCM. We'll use simple examples to show how to apply each of these methods.
5 LCM Methods
1: Listing Multiplies
Example 1: Find LCM of 8 and 6 using listing multiples method.
Step 1: The first few multiples of 6 and 8 are:
List multiples of 6: 6, 12, 18, 24, 30
List multiples of 8: 8, 16, 24, 32, 40
Step 2: LCM is the smallest number that appears in both lists:
LCM (6, 8) = 24
This method is unsuitable for numbers that are greater than 20.
2: Prime factors method
Example: Find the LCM of 8, 12 and 30.
Step 1: Prime factorizations of given numbers are:
8 = 2 · 2 · 2
12 = 2 · 2 · 3
30 = 2 · 3 · 5
Step 2: Match primes vertically
| 8 | = | 2 | · | 2 | · | 2 | ||||
| 12 | = | 2 | · | 2 | · | 3 | ||||
| 30 | = | 2 | · | 3 | · | 5 |
Step 3: Bring down numbers in each column and multiply to get LCM:
| 8 | = | 2 | · | 2 | · | 2 | ||||||
| 12 | = | 2 | · | 2 | · | 3 | ||||||
| 30 | = | 2 | · | 3 | · | 5 | ||||||
| LCM | = | 2 | · | 2 | · | 2 | · | 3 | · | 5 | = | 120 |
The Venn diagram below illustrates this solution.
Learn the prime factor method in 40 seconds.
3: Ladder method
Example: Find the LCM of 84 and 112, using Ladder method.
Step 1: Put the numbers in the division bar as follows:
| 84 | 112 |
Step 2: Divide both 84 and 112 by 2:
| 2 | 84 | 112 |
| 42 | 56 |
Step 3: Repeat Step 2 until you can no longer divide.
| 2 | 84 | 112 |
| 2 | 42 | 56 |
| 7 | 21 | 28 |
| 3 | 4 |
Step 4:LCM is the product of numbers into a L shape.
| 2 | 84 | 112 |
| 2 | 42 | 56 |
| 7 | 21 | 28 |
| 3 | 4 |
LCM = 2 · 2 · 7 · 3 · 4 = 336
4: Division method
Example: Find LCM of 18, 24 and 60 using the division method.
Step 1: Write the given numbers on a horizontal line.
| 18 | 24 | 60 |
Step 2: Divide the numbers by the smallest prime number. If a number is not divisible by two, write it down unmodified.
| 18 | 24 | 60 | |
| 2 | 9 | 12 | 30 |
| 2 | 9 | 6 | 15 |
| 2 | 9 | 3 | 15 |
Step 3:Continue dividing by larger primes 3, 5, 7… Stop when the last row contains only ones.
| 18 | 24 | 60 | |
| 2 | 9 | 12 | 30 |
| 2 | 9 | 6 | 15 |
| 2 | 9 | 3 | 15 |
| 3 | 3 | 1 | 5 |
| 3 | 1 | 1 | 5 |
| 5 | 1 | 1 | 1 |
Step 4:To get the LCM, multiply the numbers in the first column.
LCM(18,24,60) = 2·2·2·3·3·5 = 360
5: LCM formula
The formula to find LCM is:
LCM(a,b)·GCD(a,b) = a·b
Example: find LCM of 48 and 60?
Since GCD(48, 60) = 12, we have:
LCM(a,b) · GCD(a,b) = a · b
LCM(a,b) · 12 = 48 · 60
LCM(a,b) · 12 = 2880
LCM(a,b) = 2880/12
LCM(a,b) = 240
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