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Least Common Multiple — Definition, How to Find & Examples

Least Common Multiple
LCM

The smallest positive integer into which two or more integers divide evenly. For example, 24 is the LCM of 8 and 12. Sometimes the LCM is written using brackets: [8, 12] = 24.

 

 

 

See also

Greatest common factor, least common denominator

Key Formula

LCM(a,b)=∣a⋅b∣GCF(a,b)\text{LCM}(a, b) = \frac{|a \cdot b|}{\text{GCF}(a, b)}
Where:
  • a,ba, b = The two integers whose LCM you want to find
  • GCF(a,b)\text{GCF}(a, b) = The greatest common factor of a and b

Worked Example

Problem: Find the LCM of 12 and 18 using prime factorization.
Step 1: Find the prime factorization of each number.
12=22×3118=21×3212 = 2^2 \times 3^1 \qquad 18 = 2^1 \times 3^2
Step 2: For each prime factor that appears, take the highest power found in either factorization.
22 (from 12),32 (from 18)2^2 \text{ (from 12)}, \quad 3^2 \text{ (from 18)}
Step 3: Multiply these highest powers together.
LCM(12,18)=22×32=4×9=36\text{LCM}(12, 18) = 2^2 \times 3^2 = 4 \times 9 = 36
Verify: Check that 36 is divisible by both numbers: 36 ÷ 12 = 3 and 36 ÷ 18 = 2. Both divide evenly.
Answer: The LCM of 12 and 18 is 36.

Another Example

Problem: Find the LCM of 8 and 14 using the GCF formula.
Step 1: Find the GCF of 8 and 14. The factors of 8 are 1, 2, 4, 8. The factors of 14 are 1, 2, 7, 14. The greatest common factor is 2.
GCF(8,14)=2\text{GCF}(8, 14) = 2
Step 2: Apply the formula: multiply the two numbers and divide by their GCF.
LCM(8,14)=8×142=1122=56\text{LCM}(8, 14) = \frac{8 \times 14}{2} = \frac{112}{2} = 56
Verify: Check: 56 ÷ 8 = 7 and 56 ÷ 14 = 4. Both divide evenly, confirming the result.
Answer: The LCM of 8 and 14 is 56.

Frequently Asked Questions

What is the difference between LCM and GCF?
The LCM is the smallest number that is a multiple of both given numbers, while the GCF (greatest common factor) is the largest number that divides evenly into both given numbers. For example, for 12 and 18, the LCM is 36 and the GCF is 6. The LCM is always greater than or equal to the larger number, while the GCF is always less than or equal to the smaller number.
How do you find the LCM of three or more numbers?
Find the LCM of the first two numbers, then find the LCM of that result with the third number, and continue for any remaining numbers. For example, to find LCM(4, 6, 10): first LCM(4, 6) = 12, then LCM(12, 10) = 60. So LCM(4, 6, 10) = 60.

Least Common Multiple (LCM) vs. Greatest Common Factor (GCF)

The LCM finds the smallest shared multiple—the smallest number both values go into. The GCF finds the largest shared factor—the biggest number that goes into both values. They are connected by the formula LCM(a, b) × GCF(a, b) = |a × b|. The LCM is always ≥ the larger of the two numbers, while the GCF is always ≤ the smaller of the two.

Why It Matters

The LCM is essential when adding or subtracting fractions with different denominators—the least common denominator is simply the LCM of the denominators. It also appears in scheduling problems, such as finding when two events with different repeating cycles will next coincide. Beyond arithmetic, the LCM is a fundamental concept in number theory and algebra.

Common Mistakes

Mistake: Multiplying the two numbers together and assuming that is the LCM.
Correction: The product of two numbers equals the LCM only when the numbers share no common factor other than 1 (i.e., they are coprime). For example, 4 × 6 = 24, but LCM(4, 6) = 12 because they share a factor of 2. Always check for common factors or use prime factorization.
Mistake: Using the lowest power of each prime factor instead of the highest.
Correction: When using prime factorization, you must take the highest power of each prime. Taking the lowest power gives you the GCF, not the LCM. For 12 = 2² × 3 and 18 = 2 × 3², the LCM uses 2² and 3², giving 36—not 2¹ and 3¹, which would give the GCF of 6.

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