Least Common Multiple (LCM): Definition & How to Find It

Definitionnoun phrase

What is the least common multiple?

The least common multiple (LCM) is the smallest positive multiple shared by two or more numbers. The LCM of 6 and 9 is 18: it is the first number on both lists of multiples, 6, 12, 18, … and 9, 18, …. The LCM of the denominators is the least common denominator for adding fractions.

In Spanish: mínimo común múltiplo
Also called: LCM, lowest common multiple, smallest common multiple
Grade level: 6-8

Grade 5Grade 6Grade 7CCSS 6.NS.B.4GED

Least Common Multiple in a picture

Skip-count by each number and look for the first landing they share: 6 and 9 first meet at 18. Later shared landings, like 36, are common multiples but not the least.

Formal definition and notation

For positive whole numbers a and b, the least common multiple lcm(a, b) is the smallest positive whole number that is a multiple of both a and b. Every common multiple of a and b is a multiple of the LCM. In terms of prime factorizations, the LCM is the product of every prime that appears in either number, each raised to the greater of its two exponents. It is linked to the greatest common factor by lcm(a, b) × gcf(a, b) = a × b.

Symbol	Read it as	Meaning
LCM(6, 9) = 18	the LCM of 6 and 9 is 18	the usual school notation
lcm(a, b)	l c m of a and b	the same idea in lowercase; spreadsheets use the function LCM

Key facts

The LCM is never smaller than the largest of the numbers. If one number is a multiple of the other, it is the LCM: LCM(5, 15) = 15.
If two numbers share no prime factors, their LCM is their product: LCM(4, 9) = 36.
The product a × b is always a common multiple, but it is the least one only when the GCF is 1.
To add 16 + 19, use the LCM 18 as the common denominator: 318 + 218 = 518.
Repeating events line up at the LCM: lights flashing every 6 and every 9 seconds flash together every 18 seconds.

LCM from prime factors

LCM = product of every prime, each with its greatest exponent

18 = 2 × 3²: prime factorization of the first number
30 = 2 × 3 × 5: prime factorization of the second number
2 × 3² × 5 = 90: use 2, 3 and 5, each as many times as it appears in either number

Shortcut for two numbers: LCM = a × bGCF. For 18 and 30, 5406 = 90.

Least Common Multiple examples, step by step

Example 1: List the multiples

Warm-up

Find the LCM of 8 and 12.

Multiples of 12 (the larger number): 12, 24, 36, 48, …
Check each one for 8: 12 is not a multiple of 8, but 24 = 8 × 3 is.
24 is the first multiple of 12 that is also a multiple of 8.

Answer: The LCM of 8 and 12 is 24.

Example 2: Use prime factorization

Core

Find the LCM of 18 and 30.

Factor: 18 = 2 × 3² and 30 = 2 × 3 × 5.
List every prime that appears: 2, 3 and 5.
Use the greatest power of each: 2¹, 3² and 5¹.
Multiply: 2 × 9 × 5 = 90.

Answer: The LCM of 18 and 30 is 90.

Example 3: When do the buses meet?

Test-style

Bus A leaves the station every 12 minutes and Bus B every 20 minutes. Both leave at 8:00 a.m. When do they next leave at the same time?

Bus A leaves at multiples of 12 minutes after 8:00; Bus B at multiples of 20.
They match at a common multiple, and the next match is the least one: LCM(12, 20).
12 = 2² × 3 and 20 = 2² × 5, so the LCM is 2² × 3 × 5 = 60 minutes.
Trap: 12 × 20 = 240 minutes is a common multiple, but not the first one.

Answer: They next leave together 60 minutes later, at 9:00 a.m.

Common mistakes

✗ MistakeThe LCM of 6 and 8 is 2.

✓ InsteadThe LCM of 6 and 8 is 24. (2 is their GCF.)

The least common multiple is a multiple, not a factor. It can never be smaller than the larger number.

✗ MistakeLCM(6, 8) = 6 × 8 = 48.

✓ Instead48 is a common multiple, but 24 is the least: 24 = 6 × 4 = 8 × 3.

Multiplying works only when the numbers share no factor. 6 and 8 share a 2, so the product counts it twice.

✗ Mistake12 = 2² × 3 and 18 = 2 × 3², so the LCM is 2 × 3 = 6.

✓ InsteadTake each prime with its greatest exponent: 2² × 3² = 36.

Keeping only the shared primes with the smaller exponents gives the GCF. The LCM needs enough of each prime to build both numbers.

✗ MistakeLCM(5, 15) = 75

✓ InsteadLCM(5, 15) = 15

15 is already a multiple of 5 (5 × 3), so it is the smallest number both divide into.

Not to be confused with

Least Common Multiplevs.Greatest Common Factor

The GCF is the largest number that divides both; the LCM is the smallest number that both divide. For 6 and 9: GCF = 3, LCM = 18.

Least Common Multiplevs.Common multiple

Any number that is a multiple of both is a common multiple: 18, 36, 54, … for 6 and 9. The LCM is the smallest of them.

Least Common Multiplevs.Least common denominator

The least common denominator is the LCM of the denominators of some fractions. For 16 and 19 it is 18, the LCM of 6 and 9.

Quick check: 3 questions

Question 1 of 3

What is the LCM of 4 and 10?

Show answer and explanation

Answer: C. Multiples of 10: 10, 20, … and 20 is also a multiple of 4. 40 is the product (a common multiple, but not the least), 2 is the GCF, and 14 is the sum.

Question 2 of 3

What is the LCM of 7 and 21?

Show answer and explanation

Answer: A. 21 is a multiple of 7 (7 × 3), so 21 is the least common multiple. 147 is the product, 7 is the GCF, and 3 is the quotient 21 ÷ 7.

Question 3 of 3

Use the LCM of the denominators to add 16 + 38.

Show answer and explanation

Answer: B. LCM(6, 8) = 24, so 424 + 924 = 1324. 414 adds tops and bottoms, and 1348 uses 48 as the denominator without rescaling the numerators correctly.

Least Common Multiple: frequently asked questions

How do you find the LCM of three numbers?

Factor all three into primes, then multiply every prime that appears, each with its greatest exponent. For 4, 6 and 10: 4 = 2², 6 = 2 × 3, 10 = 2 × 5, so the LCM is 2² × 3 × 5 = 60. You can also find the LCM of two numbers, then the LCM of that result and the third.

What is the LCM used for?

The LCM of the denominators is the least common denominator for adding or subtracting fractions, which keeps the numbers small. It also answers “when will these line up again?” questions: two lights blinking every 4 and 6 seconds blink together every 12 seconds, and items sold in packs of 8 and 12 match at 24.

How are the LCM and GCF related?

For two positive whole numbers, the LCM times the GCF equals the product of the numbers. For 12 and 18: GCF = 6, LCM = 36, and 6 × 36 = 216 = 12 × 18. So if you know the GCF, you can get the LCM by dividing the product by it: 216 ÷ 6 = 36.

Practice and tools

Math practicePractice on Effortless Math GCF and LCM CalculatorFind the LCM of two or more numbers with steps Least Common Denominator CalculatorUse the LCM to get a common denominator Adding Fractions with Unlike DenominatorsSee the LCM at work when adding fractions

Sources

Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 6.NS.B.4

Cite this entry

Effortless Math Team. “Least Common Multiple.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/least-common-multiple/

Flashcard

TermLeast Common MultipleLCM of 6 and 9

DefinitionThe smallest positive number that is a multiple of each number. 6, 12, 18 and 9, 18: LCM(6, 9) = 18.

Study the full deck

Browse all terms in the Math Dictionary · More number system terms · Random term

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Least Common Multiple

What is the least common multiple?

Least Common Multiple in a picture

Formal definition and notation

Key facts

LCM from prime factors

Least Common Multiple examples, step by step

Example 1: List the multiples

Example 2: Use prime factorization

Example 3: When do the buses meet?

Common mistakes

Not to be confused with

Quick check: 3 questions

Least Common Multiple: frequently asked questions

How do you find the LCM of three numbers?

What is the LCM used for?

How are the LCM and GCF related?

Related terms

Practice and tools

Sources

Flashcard

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