Enter 2 or more numbers separated by commas (,). You can find the LCM of 3 or more numbers at once, and the steps with prime factorization and the greatest common factor (GCF) are shown too.
Enter 2 or more positive whole numbers separated by commas (for example, 4, 6). 0, decimals and negative numbers cannot be used.
Result
Enter your numbers separated by commas in the field on the left and press "Calculate". The result will appear here.
Table of Contents
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What you can do on this page
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What is this calculation used for?
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How to Use
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Formula
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Symbols and terms
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Good to know before you start
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How to calculate it in Excel
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How to calculate it in Google Sheets
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How to calculate it in Python
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How to write it in LaTeX and other math languages (copy and paste)
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How to have ChatGPT do the calculation
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DataChef Features
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Related Features
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NumberChef Calculators List
What you can do on this page
- Enter 2 or more numbers separated by commas (,) and get the least common multiple (LCM) on the spot
- Not only "What is the LCM of 4 and 6?" but also 3 or more numbers at once, such as "What is the LCM of 12, 18 and 30?"
- The steps show the prime factorization of each number, so you see why the answer is what it is, not just the answer
- The greatest common factor (GCF) is shown at the same time for reference
- A plain-language explanation of the formulas and copy-and-paste formulas for Excel, Google Sheets and Python are all on this page
Only positive whole numbers (1, 2, 3, …) can be entered. The least common multiple is defined for positive whole numbers, so 0, decimals and negative numbers cannot be used.
What is this calculation used for?
Finding a common denominator to add or subtract fractions (its biggest use)
To calculate \(\dfrac{1}{4} + \dfrac{1}{6}\), the denominators 4 and 6 have to be made the same. The least common denominator is the LCM, \(\mathrm{lcm}(4,6) = 12\), so the sum is \(\dfrac{3}{12} + \dfrac{2}{12} = \dfrac{5}{12}\).
Multiplying the denominators to get 24 also works, but the LCM keeps the numbers small and makes simplifying easier at the end. From elementary school math to everyday work, this is where the LCM is used most.
Finding when buses or trains next leave at the same time
Two bus routes leave the same stop together, one every 12 minutes and one every 18 minutes. They next leave together after \(\mathrm{lcm}(12,18) = 36\) minutes.
The moment when two things that repeat on different cycles line up is given by the LCM. It is a classic word problem in school math, and a practical tool for thinking about timetables and other schedules.
Buying hot dogs and buns so none are left over
Hot dogs come in packs of 10 and buns in packs of 8. To have exactly the same number of each, the fewest you can buy is \(\mathrm{lcm}(10,8) = 40\) of each, which is 4 packs of hot dogs and 5 packs of buns.
Matching things that come in different pack sizes happens everywhere, from shopping for a cookout to ordering parts for a factory.
Designing gears (machines and clocks)
A gear with 12 teeth turns with a gear with 18 teeth. The two teeth that touched at the start meet again after \(\mathrm{lcm}(12,18) = 36\) teeth have passed (3 turns of the small gear and 2 turns of the large one).
In machine design, the tooth counts are often chosen to be relatively prime to make the LCM large, so the same pairs of teeth do not keep hitting each other and wear unevenly.
When 13-year and 17-year cicadas come out together (nature)
In the eastern US, some broods of periodical cicadas come out of the ground every 13 years and others every 17 years. A 13-year brood and a 17-year brood that come out in the same year will next do so after \(\mathrm{lcm}(13,17) = 221\) years. This happened in 2024, for the first time since 1803.
Because 13 and 17 are both prime, their LCM is simply \(13 \times 17\). How long it takes for things with different cycles to line up again is exactly what the LCM tells you.
Formula
LCM of two numbers (using the GCF)
Standard notation
(the usual math form)
\(\mathrm{lcm}(a,\,b)\)
\(=\)
\(a\)
\(\times\)
\(b\)
\(\div\)
\(\gcd(a,\,b)\)
In words
(symbols replaced with words)
④
\(\mathrm{lcm}(a,b)\): least common multiple
\(=\)
①
\(a\): first number
\(\times\)
②
\(b\): second number
\(\div\)
③
\(\gcd(a,b)\): greatest common factor
The formula in words
①
Multiply the
\(a\): first number
②
by the
\(b\): second number
③
divide by the
\(\gcd(a,b)\): greatest common factor
④
and you get the
\(\mathrm{lcm}(a,b)\): least common multiple
Quick example
The LCM of 4 and 6 (their GCF is 2) is
LCM of 4 and 6
\(=\)
first number (4)
\(\times\)
second number (6)
\(\div\)
GCF (2)
\(4 \times 6 \div 2 = 24 \div 2 = 12\)
Key idea
If you just multiply the two numbers, the part they have in common (the greatest common factor) is counted twice. Dividing by it once removes the extra copy, and what is left is the least common multiple.
When the GCF is 1 (the numbers are relatively prime), dividing by 1 changes nothing, so the LCM is simply the product \(\mathrm{lcm}(a,b) = a \times b\) (for 8 and 9, \(8 \times 9 = 72\)).
Using prime factorization (works for 3 or more numbers)
Standard notation
(the usual math form)
\(L\)
\(=\)
\(p_1^{\,a_1} \times p_2^{\,a_2} \times \cdots \times p_k^{\,a_k}\)
In words
(symbols replaced with words)
②
\(L\): least common multiple
\(=\)
①
product of the highest power of each prime that appears
The formula in words
①
Write each number as a product of primes and take the
product of the highest power of each prime that appears
②
to get the
\(L\): least common multiple
Quick example
For the LCM of 12, 18 and 30: the prime factorizations are \(12 = 2^2 \times 3\), \(18 = 2 \times 3^2\) and \(30 = 2 \times 3 \times 5\), and the highest powers of the primes 2, 3 and 5 are \(2^2\), \(3^2\) and \(5\), so
LCM of 12, 18 and 30, \(L\)
\(=\)
product of the highest powers \(2^2 \times 3^2 \times 5\)
\(2^2 \times 3^2 \times 5 = 4 \times 9 \times 5 = 180\)
Key idea
Why the highest exponent? Because the least common multiple has to be a multiple of every one of the numbers. For example, to be a multiple of \(12 = 2^2 \times 3\), it needs two 2s, so the exponent of 2 is set to 2, the largest one that appears. For each prime, match the number that needs it most: that is the whole idea of this method.
The steps shown by the calculator on this page also use this prime factorization method.
LCM of 3 or more numbers (two at a time)
Standard notation
(the usual math form)
\(\mathrm{lcm}(a,\,b,\,c)\)
\(=\)
\(\mathrm{lcm}(m,\,c)\)
\(m\)
\(=\)
\(\mathrm{lcm}(a,\,b)\)
In words
(symbols replaced with words)
③
LCM of all three numbers
\(=\)
②
LCM of \(m\) and \(c\)
①
\(m\): LCM of \(a\) and \(b\)
\(=\)
\(\mathrm{lcm}(a,b)\): LCM of \(a\) and \(b\)
The formula in words
①
First, find the
\(m\): LCM of \(a\) and \(b\)
②
then find the
LCM of \(m\) and \(c\)
③
and that is the
LCM of all three numbers
(for 4 or more numbers, just keep repeating the same step)
Quick example
For the LCM of 12, 18 and 30, first find the LCM of 12 and 18 (36), then the LCM of that 36 and 30:
LCM of 12, 18 and 30
\(=\)
LCM of "the LCM of 12 and 18 (36)" and 30
\(\mathrm{lcm}(12,\,18) = 36\)
\(\mathrm{lcm}(36,\,30) = 180\)
Key idea
You can start with any two of the numbers and the answer is the same (the order does not change the result). The calculator on this page also uses this two-at-a-time method to handle any number of inputs.
The basic way to find the least common multiple is to write each number as a product of primes, pick the highest power of each prime that appears, and multiply them. For two numbers, the shortcut is "the product of the two numbers divided by their GCF". For 3 or more numbers, find the LCM two at a time.
Symbols and terms
Symbols
| \(\mathrm{lcm}(a,b)\) | L C M of a and b | The least common multiple of \(a\) and \(b\). It comes from the first letters of "least common multiple" and is also written in capitals, LCM. |
| \(\gcd(a,b)\) | G C D of a and b | The greatest common factor of \(a\) and \(b\). The symbol comes from "greatest common divisor", another name for the same number. US schools usually call it the GCF (greatest common factor). |
| \(2^2\) | two squared | A power: the small raised number (the exponent) tells how many times the same number is multiplied. \(2^2 = 2 \times 2 = 4\) and \(3^2 = 3 \times 3 = 9\). |
| \(p_1, p_2, \ldots, p_k\) | p sub 1, p sub 2 | The primes that appear in the prime factorizations, from smallest to largest. The small lowered number (the subscript) tells which prime it is in the list. |
| \(a_1, a_2, \ldots, a_k\) | a sub 1, a sub 2 | For each prime, the largest exponent (number of times it is multiplied) that appears in any of the prime factorizations. |
| \(m\) | m | The LCM of the first two numbers, found along the way. It is used when finding the LCM of 3 or more numbers two at a time. |
Terms
| multiple | A number times 1, 2, 3 and so on. The multiples of 6 are 6, 12, 18, 24, … |
| common multiple | A multiple shared by two or more numbers. The common multiples of 4 and 6 are 12, 24, 36, … and they go on forever. |
| least common multiple (LCM) | The smallest of the common multiples. This is the value this page finds. |
| factor | A whole number that divides a number evenly. The factors of 12 are 1, 2, 3, 4, 6 and 12. |
| greatest common factor (GCF) | The largest factor shared by two or more numbers. It is also called the greatest common divisor (GCD). It often goes hand in hand with the LCM, so this page shows it for reference. |
| prime number | A whole number of 2 or more whose only factors are 1 and itself (2, 3, 5, 7, 11, …). 1 is not a prime number. |
| prime factorization | Writing a whole number as a product of prime numbers only. Example - \(12 = 2 \times 2 \times 3 = 2^2 \times 3\). |
| relatively prime | When the GCF of two numbers is 1 (they have no prime factor in common). The LCM of two relatively prime numbers is just their product. |
| least common denominator (LCD) | The least common multiple of the denominators. To add or subtract fractions with different denominators, you rewrite them with the same denominator, and the LCD is the simplest one to use. This is where the LCM is used most often. |
Good to know before you start
Here is what helps you use the calculation on this page with real understanding, not just by pressing the button.
| Multiplication facts and division (Grade 3) |
|
| Multiples and factors (Grades 4–6) |
|
| Common denominators (Grade 5) |
|
| Prime numbers and prime factorization (Grades 4–6) |
|
| Exponents (Grade 6) |
|
How to calculate it in Excel
Copy the whole table below and paste it into cell A1 in Excel. It works as is.
Table to find the LCM of two numbers (using the GCF)
| First number a | 4 |
| Second number b | 6 |
| LCM lcm(a, b) | =B1*B2/GCD(B1,B2) |
Table to find the LCM of 3 or more numbers
| First number | 12 |
| Second number | 18 |
| Third number | 30 |
| LCM | =LCM(B1,B2,B3) |
Table to find the LCM two at a time
| First number | 12 |
| Second number | 18 |
| Third number | 30 |
| First, LCM of the first two, m | =LCM(B1,B2) |
| LCM of all three | =LCM(B4,B3) |
After pasting, the upper cells in column B are your inputs and the last row is calculated automatically.
"GCD" finds the greatest common factor and "LCM" finds the least common multiple in Excel. The LCM function does the prime factorization of formula 2 for you.
The first table shows 12 in B3, and the second table shows 180 in B4. Just replace the numbers with your own (the LCM function takes more numbers, such as "=LCM(B1,B2,B3,B4)").
"GCD" finds the greatest common factor and "LCM" finds the least common multiple in Excel. The LCM function does the prime factorization of formula 2 for you.
The first table shows 12 in B3, and the second table shows 180 in B4. Just replace the numbers with your own (the LCM function takes more numbers, such as "=LCM(B1,B2,B3,B4)").
How to calculate it in Google Sheets
Copy the whole table below and paste it into cell A1 in Google Sheets. It works as is.
Table to find the LCM of two numbers (using the GCF)
| First number a | 4 |
| Second number b | 6 |
| LCM lcm(a, b) | =B1*B2/GCD(B1,B2) |
Table to find the LCM of 3 or more numbers
| First number | 12 |
| Second number | 18 |
| Third number | 30 |
| LCM | =LCM(B1,B2,B3) |
Table to find the LCM two at a time
| First number | 12 |
| Second number | 18 |
| Third number | 30 |
| First, LCM of the first two, m | =LCM(B1,B2) |
| LCM of all three | =LCM(B4,B3) |
Google Sheets has the same LCM and GCD functions, so the formulas from Excel work as is. Copy the whole table, paste it into cell A1, and replace the numbers with your own.
How to calculate it in Python
import math
numbers = [330, 75, 450, 225] # numbers to find the LCM of (2 or more)
least_common_multiple = math.lcm(*numbers)
greatest_common_divisor = math.gcd(*numbers)
print(f"Least common multiple (LCM): {least_common_multiple}")
print(f"(For reference) greatest common factor (GCF): {greatest_common_divisor}")
Runs with the standard library only (math.lcm needs Python 3.9 or later). "*numbers" passes the items of the list to the function one by one, so it works with any number of values. Change the numbers in the list at the top and run it.
How to write it in LaTeX and other math languages (copy and paste)
LCM of two numbers (using the GCF)
lcm(a, b) = a × b ÷ gcd(a, b)
\mathrm{lcm}(a, b) = \dfrac{a \times b}{\gcd(a, b)}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>lcm</mi><mo>⁡</mo>
<mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow>
<mo>=</mo>
<mfrac>
<mrow><mi>a</mi><mo>×</mo><mi>b</mi></mrow>
<mrow><mi>gcd</mi><mo>⁡</mo><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow></mrow>
</mfrac>
</mrow>
</math>
lcm(a, b) = (a xx b) / (gcd(a, b))
a*b/GCD[a, b]
L := a*b/igcd(a, b);
L = a*b/gcd(a, b);
lcm(a, b) = (a × b)/gcd(a, b)
Using prime factorization (works for 3 or more numbers)
L = p_1^{a_1} \times p_2^{a_2} \times \cdots \times p_k^{a_k}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>L</mi>
<mo>=</mo>
<msubsup><mi>p</mi><mn>1</mn><msub><mi>a</mi><mn>1</mn></msub></msubsup>
<mo>×</mo>
<msubsup><mi>p</mi><mn>2</mn><msub><mi>a</mi><mn>2</mn></msub></msubsup>
<mo>×</mo>
<mo>⋯</mo>
<mo>×</mo>
<msubsup><mi>p</mi><mi>k</mi><msub><mi>a</mi><mi>k</mi></msub></msubsup>
</mrow>
</math>
L = p_1^(a_1) xx p_2^(a_2) xx cdots xx p_k^(a_k)
Product[p[i]^a[i], {i, 1, k}]
L := product(p[i]^a[i], i = 1 .. k);
L = prod(p.^a);
L = p_1^(a_1) × p_2^(a_2) × ⋯ × p_k^(a_k)
LCM of 3 or more numbers (two at a time)
lcm(a, b, c) = lcm(lcm(a, b), c)
\mathrm{lcm}(a, b, c) = \mathrm{lcm}(\mathrm{lcm}(a, b),\ c)
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>lcm</mi><mo>⁡</mo>
<mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>)</mo></mrow>
<mo>=</mo>
<mi>lcm</mi><mo>⁡</mo>
<mrow><mo>(</mo><mi>lcm</mi><mo>⁡</mo><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow><mo>,</mo><mi>c</mi><mo>)</mo></mrow>
</mrow>
</math>
lcm(a, b, c) = lcm(lcm(a, b), c)
LCM[LCM[a, b], c]
L := ilcm(ilcm(a, b), c);
L = lcm(lcm(a, b), c);
lcm(a, b, c) = lcm(lcm(a, b), c)
How to have ChatGPT do the calculation
You are a calculation assistant for whole numbers. Do the following calculation by actually running Python code, and base your answer only on the numbers from the execution result (do not answer by mental math or guessing). For the four numbers 330, 75, 450 and 225, find each of the following: 1. The least common multiple (LCM) 2. The greatest common factor (GCF) 3. The prime factorization of each number Show the formulas you used and the numbers from the execution result.
How to Use
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1Enter your numbersType the numbers you want to calculate with into the input fields
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2CalculatePress the "Calculate" button
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3Check the resultThe result appears on the spot. The same page also explains the idea behind the calculation and the formula
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