Listing Multiples
Write out multiples of each number and circle the smallest one they share. The most intuitive way to build understanding.
Free calculator for students & teachers
Find the Least Common Multiple,
Learn more
Find the Least Common Multiple,
instantly.
A free least common multiple calculator (also the lowest common denominator when you're adding fractions) with step-by-step explanations. Enter up to 10 numbers and see three ways to reach the answer.
Example preview
LCM =
60
Enter numbers above and press Calculate to see your answer.
Step-by-step solution
Example with 12 and 15:
- Multiples of 12: 12, 24, 36, 48, 60
- Multiples of 15: 15, 30, 45, 60
The smallest common multiple is 60.
The concept
What is the least common multiple?
In mathematics, the Least Common Multiple (LCM) of two or more whole numbers is the smallest positive common multiple that every one of them divides into evenly. It is also called the lowest common multiple, and when you apply it to the denominators of fractions it becomes the least common denominator (LCD).
For example, the LCM of 4 and 6 is 12, the smallest number that both 4 and 6 divide with no remainder. The same 12 is the least common denominator when you're adding 1⁄4 and 1⁄6.
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Adding fractions with different denominators.
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Scheduling events that repeat on different cycles.
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Music rhythms that line up after so many beats.
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Machines & signals with different periods.
Three approaches
Find the LCM, your way.
Three proven methods for different problems. Pick the one that matches your numbers.
Prime Factorization
Break each number into primes, take the highest power of each, then multiply. Shows why the LCM works, not just what it is.
GCD Formula
Use the Euclidean shortcut: (a × b) ÷ GCD(a, b). Works instantly on huge numbers where listing would take forever.
Cheat sheet
LCM formulas, at a glance.
The three ways to compute LCM, distilled. Save, bookmark, or screenshot for homework.
LCM(a, b)
=
|a · b|
GCD(a, b)
The fastest way for two numbers. Works with any pair of positive integers.
LCM = p1max · p2max · p3max …
Finding LCM using prime factorization: list every prime that appears anywhere, then raise it to its highest power across all numbers. This is how you handle LCM with exponents.
LCM(a, b, c)
=
LCM(LCM(a, b), c)
Chain the two-number formula. LCM is associative, so the order you pair them doesn't matter.
LCM(a, b) × GCD(a, b) = a × b
LCM and GCD are two sides of the same coin. If you know one, the other is a division away.
LCM in the wild
Where LCM actually matters.
Beyond the textbook, the least common multiple pops up in biology, music, and machinery.
The Cicada Puzzle
Two broods of cicadas emerge every 13 and 17 years. How often do they meet?
Music Rhythms
A drummer plays every 6 beats, a pianist every 8. When do they land together?
Gears & Cycles
Two gears with 12 and 18 teeth. How many turns until they reset to the start?
LCM chart · look-up table
Common LCM values.
The least common multiple and greatest common divisor for the pairs students search for the most. Each row doubles as a lowest common denominator look-up for fractions — tap to auto-fill the calculator with full step-by-step working.
| Pair | LCM | GCD | |
|---|---|---|---|
| LCM of 2 and 3 (common denominator of 2 and 3) | 6 | 1 | Try → |
| LCM of 3 and 4 | 12 | 1 | Try → |
| LCM of 3 and 6 (common denominator of 3 and 6) | 6 | 3 | Try → |
| LCM of 3 and 8 (common multiple of 3 and 8) | 24 | 1 | Try → |
| LCM of 4 and 5 | 20 | 1 | Try → |
| LCM of 4 and 6 | 12 | 2 | Try → |
| LCM of 4 and 10 | 20 | 2 | Try → |
| LCM of 5 and 6 | 30 | 1 | Try → |
| LCM of 6 and 8 (common multiples of 6 and 8) | 24 | 2 | Try → |
| LCM of 6 and 9 | 18 | 3 | Try → |
| LCM of 8 and 12 (common multiples of 8 and 12) | 24 | 4 | Try → |
| LCM of 9 and 12 | 36 | 3 | Try → |
| LCM of 10 and 15 | 30 | 5 | Try → |
| LCM of 12 and 15 | 60 | 3 | Try → |
| LCM of 12 and 16 | 48 | 4 | Try → |
| LCM of 12 and 18 (common multiples of 12 and 18) | 36 | 6 | Try → |
| LCM of 14 and 21 | 42 | 7 | Try → |
| LCM of 15 and 20 | 60 | 5 | Try → |
| LCM of 18 and 24 | 72 | 6 | Try → |
| LCM of 20 and 30 | 60 | 10 | Try → |
| LCM of 24 and 36 | 72 | 12 | Try → |
Can't find your pair? Use the calculator above for any two or more numbers.
GCF and LCM, side by side
LCM vs GCD.
The greatest common factor and least common multiple are the two most-confused ideas in number theory. Here's the side-by-side you wish your textbook had — GCD, GCF, and HCF all mean the same thing.
LCM Least Common Multiple
- What it is
- The smallest number that every input divides into evenly.
- Key property
- Always ≥ the largest input.
- Example
- LCM(4, 6) = 12. Both 4 and 6 divide 12.
- Typical use
- Adding fractions with different denominators. Syncing repeating events.
GCD Greatest Common Divisor
- What it is
- The largest number that divides every input evenly.
- Key property
- Always ≤ the smallest input.
- Example
- GCD(4, 6) = 2. The biggest number that divides both.
- Typical use
- Simplifying fractions. Reducing ratios to lowest terms.
Linked by one elegant identity:
LCM(a, b) × GCD(a, b) = a × b
Three quick steps
How to use this calculator.
-
01
Type your numbers
Enter up to 10 positive whole numbers, separated by commas, spaces, or new lines.
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02
Pick a method
Listing multiples, prime factorization, or the GCD formula. Pick whichever you want to learn.
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03
Get step-by-step
See the full working out: colored prime factors, pair-wise GCD, or a visual multiples list.
Your turn
Practice problems.
Try each one in your head or on paper, then click "Reveal" to check your answer. No score, no signup, just reps.
01 Find the LCM of 6 and 8.
Answer: 24
Prime factors: 6 = 2·3, 8 = 23. Take the highest power of each prime: 23 · 3 = 8 · 3 = 24.
02 Find the LCM of 9 and 12.
Answer: 36
9 = 32, 12 = 22·3. Highest powers: 22 · 32 = 4 · 9 = 36.
03 Find the LCM of 10, 15, and 20.
Answer: 60
10 = 2·5, 15 = 3·5, 20 = 22·5. Highest powers: 22 · 3 · 5 = 4 · 3 · 5 = 60.
04 Find the LCM of 7 and 11.
Answer: 77
7 and 11 are both prime and coprime (GCD = 1). When that happens, LCM is simply their product: 7 · 11 = 77.
05 Find the LCM of 14 and 35.
Answer: 70
14 = 2·7, 35 = 5·7. Highest powers: 2 · 5 · 7 = 70. Notice that 7 appears in both, but you still only count it once.
06 Find the LCM of 24 and 36.
Answer: 72
24 = 23·3, 36 = 22·32. Highest powers: 23 · 32 = 8 · 9 = 72. Or use the formula: (24 · 36) ÷ GCD(24, 36) = 864 ÷ 12 = 72.
Watch out
Common mistakes.
Six traps that trip up most students. Skim these before your next homework.
Multiplying all the numbers
Wrong LCM(4, 6) = 4 × 6 = 24.
Right LCM(4, 6) = 12. Multiplying only works when the numbers share no common factors (i.e. GCD = 1).
Taking the lowest power
Wrong For 8 = 23 and 12 = 22·3, taking 22 gives 12.
Right LCM uses the highest power of each prime: 23·3 = 24. Lowest powers belong to GCD, not LCM.
Forgetting a prime
Wrong LCM(6, 35): only counting 5 and 7 because 6 isn't prime.
Right First factor: 6 = 2·3, 35 = 5·7. Every prime that appears anywhere must be included. LCM = 2·3·5·7 = 210.
Confusing LCM with GCD
Wrong Answering "2" for LCM of 4 and 6.
Right 2 is the GCD. The LCM is 12. Remember: LCM is always ≥ largest input, GCD is always ≤ smallest input.
Using zero or negatives
Wrong LCM(0, 5) = 0.
Right LCM is only defined for positive integers. Zero has every number as a multiple, so there is no "least". Drop zeros and use absolute values for negatives.
Listing multiples forever
Wrong For LCM(7, 12), listing 7, 14, 21, 28… and giving up.
Right For large or coprime pairs, switch methods. Prime factorization or the GCD formula get you there faster. LCM(7, 12) = 84.
Key terms
LCM glossary.
The vocabulary you will see in any LCM problem, explained in one line each.
- Multiple
- A number you get by multiplying a given number by an integer. Multiples of 4 are 4, 8, 12, 16, 20, …
- Factor (divisor)
- A number that divides another evenly with no remainder. Factors of 12 are 1, 2, 3, 4, 6, 12.
- Prime number
- A whole number greater than 1 with no divisors other than 1 and itself. 2, 3, 5, 7, 11, 13, …
- Prime factorization
- Writing a number as a product of primes. 60 = 22 · 3 · 5.
- Common multiple
- A number that is a multiple of two or more given numbers. 24 is a common multiple of 4 and 6.
- Least Common Multiple (LCM)
- The smallest positive common multiple of two or more numbers. Also called the lowest common multiple or smallest common multiple.
- Least Common Denominator (LCD)
- The LCM applied to the denominators of fractions. Used to add or subtract fractions with unlike denominators. LCD of 1⁄4 and 1⁄6 is 12.
- Greatest Common Divisor (GCD / GCF / HCF)
- The largest number that divides every given number evenly. Also called greatest common factor or highest common factor.
- Coprime
- Two numbers with GCD = 1. They share no prime factors. 8 and 15 are coprime.
- Integer
- A whole number: positive, negative, or zero. LCM is defined on positive integers only.
Questions?
Frequently asked.
What is the least common multiple in math?
The least common multiple (LCM) of two or more integers is the smallest positive integer that is divisible by every one of them. It's also called the lowest common multiple or the smallest common multiple.
How do I find the LCM by hand?
Use one of three methods: listing multiples, finding LCM using prime factorization, or the formula LCM(a,b) = (a × b) ÷ GCD(a,b). Our calculator shows the full step-by-step working for whichever method you pick.
Is the LCM the same as the lowest common denominator (LCD)?
Yes. When you apply the least common multiple to the denominators of fractions, it's called the least common denominator or lowest common denominator. For example, the LCD of 1/4 + 1/6 is 12 — the LCM of 4 and 6. So this tool doubles as an LCD calculator.
What's the difference between LCM and GCD / GCF?
GCD (also called GCF, greatest common factor, or HCF, highest common factor) is the largest number that divides every input. LCM is the smallest number that every input divides into. They are linked by the identity LCM(a,b) × GCD(a,b) = a × b.
How do I handle LCM with exponents?
Factor each number into primes and keep the highest power of every prime. For example, LCM of 8 = 23 and 12 = 22·3 uses 23·3 = 24. Never take the lowest power — that's for GCD.
What is the LCM of 3 and 4?
The LCM of 3 and 4 is 12. Since 3 and 4 share no common factors (GCD = 1), the LCM is just their product: 3 × 4 = 12.
What is the LCM of 8 and 12?
The LCM of 8 and 12 is 24. In primes, 8 = 23 and 12 = 22·3, so LCM = 23·3 = 24.
Can the LCM be smaller than the input numbers?
No. The least common multiple is always at least as large as the biggest input number.
Is "least common factor" a real thing?
Not really. People sometimes say "least common factor" when they mean the greatest common factor (GCF) or the least common multiple (LCM). The smallest factor every integer shares is just 1, which isn't useful — so always double-check whether the problem wants the LCM or the GCF/GCD.