This calculator computes LCM of two or more numbers using five methods: the division method, listing multiples, prime factors, ladder method and the LCM formula. For each method, the calculator shows a detailed step-by-step explanation.
The least common multiple (LCM) is the smallest number that is divisible by all given numbers. For example, LCM(8, 6) = 24, because 24 is divisible by 8 and it is divisible by 6, and it is the smallest such number.
Example: find LCM of 8 and 6 by listing multiples.
Step 1: The first few multiples of 6 and 8 are:
Multiples of 6: 6, 12, 18, 24, 30
Multiples of 8: 8, 16, 24, 32, 40
Step 2: LCM is the smallest numbers that appears in both lists:
LCM (6, 8) = 24
Note:This method is unsuitable for numbers that are greater than 20.
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 |
Example: Find the LCM of 84 and 112, using Ladder method.
Step 1: Place the numbers inside the division bar:
84 | 112 |
Step 2: Divide both numbers 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 a product of numbers into L shape.
2 | 84 | 112 |
2 | 42 | 56 |
7 | 21 | 28 |
3 | 4 |
LCM = 2 · 2 · 7 · 3 · 4 = 336
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 numbers by the smallest prime number. If any number is not divisible by 2 write it down unchanged.
18 | 24 | 60 | |
2 | 9 | 12 | 30 |
2 | 9 | 6 | 15 |
2 | 9 | 3 | 15 |
Step 3:Continue dividing by prime numbers 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:Multiply the numbers in the first column to get LCM
LCM(18, 24, 60) = 2 · 2 · 2 · 3 · 3 · 5 = 360
Example: find LCM of 48 and 60?
In this section we use formula
$$ \text{LCM(a,b)} = \dfrac{ a \cdot b }{ \text{GCD(a,b)} } $$Since GCD(48, 60) = 12, we have:
$$ \text{LCM(48, 60)} = \dfrac{ 48 \cdot 60 }{ \text{GCD(48, 60)} } = \dfrac{2880}{12} = 240$$