LCM Calculator — Least Common Multiple
Calculate the least common multiple (LCM) of two or more numbers. Also shows the prime factorization of each number. See also GCF Calculator and Fraction Calculator.
Enter at least 2 positive integers separated by commas
How to Find the LCM
The least common multiple (LCM) of two or more numbers is the smallest positive integer that is divisible by all of them. There are several methods to find the LCM: prime factorization (take the highest power of each prime), the formula LCM(a,b) = |a×b| / GCD(a,b), or listing multiples until you find a common one. This calculator uses the GCD method for efficiency and also shows the prime factorization of each input number.
LCM Formula
LCM(a, b) = |a × b| / GCD(a, b)
Using prime factorization:
LCM = product of highest powers of all primes
For multiple numbers:
LCM(a, b, c) = LCM(LCM(a, b), c)
Example
LCM(12, 18) = ?
12 = 2² × 3
18 = 2 × 3²
LCM = 2² × 3² = 4 × 9
= 36
Frequently Asked Questions
What is the difference between LCM and GCF?
LCM (Least Common Multiple) is the smallest number divisible by all given numbers. GCF (Greatest Common Factor) is the largest number that divides all given numbers. They are related: LCM(a,b) × GCF(a,b) = a × b.
When do you use LCM?
LCM is used when adding or subtracting fractions (finding a common denominator), scheduling problems (when events with different cycles align), and in number theory. For example, to add 1/4 + 1/6, you need LCM(4,6) = 12 as the common denominator.
Can LCM be calculated for more than 2 numbers?
Yes. Calculate LCM of the first two numbers, then find the LCM of that result with the third number, and so on. This calculator supports any number of inputs.
Is LCM always greater than or equal to the input numbers?
Yes. The LCM must be divisible by each input number, so it is always at least as large as the largest input. If one number is a multiple of the other, the LCM equals the larger number.
Solved Examples — LCM
Example: Find the LCM of 15 and 20
Solution:
Step 1: Prime factorization: 15 = 3 × 5, 20 = 2² × 5
Step 2: Take highest power of each prime: 2², 3, 5
Step 3: LCM = 4 × 3 × 5 = 60
Answer: LCM(15, 20) = 60
Example: Two buses depart at the same time. Bus A comes every 12 minutes, Bus B every 18 minutes. When will both arrive together next?
Solution:
Step 1: Find LCM(12, 18)
Step 2: 12 = 2² × 3, 18 = 2 × 3²
Step 3: LCM = 2² × 3² = 4 × 9 = 36
Answer: Both arrive together in 36 minutes
Example: Find the LCM of 8, 12, and 15
Solution:
Step 1: 8 = 2³, 12 = 2² × 3, 15 = 3 × 5
Step 2: Take highest power of each prime: 2³, 3, 5
Step 3: LCM = 8 × 3 × 5 = 120
Answer: LCM(8, 12, 15) = 120
Practice Questions
Try these on your own:
- Find LCM(24, 36) (Answer: 72)
- Find LCM(7, 13) (Answer: 91)
- Find LCM(6, 8, 12) (Answer: 24)
- What is the LCD for adding 1/6 + 1/8 + 1/9? (Answer: 72)
- Three traffic lights cycle every 45, 60, and 75 seconds. When do they all turn green together? (Answer: 900 seconds = 15 minutes)
- If LCM(a, 15) = 60, and a is less than 60, what are possible values of a? (Answer: 4, 12, 20, 60)
Common Mistakes to Avoid
The most common mistake is confusing LCM with GCF. LCM is always greater than or equal to the largest input number, while GCF is always less than or equal to the smallest input. Another error is multiplying the numbers together and calling it the LCM — while a × b is always a common multiple, it is only the LEAST common multiple when the numbers share no common factors (are coprime). For example, LCM(4, 6) = 12, not 24. When using prime factorization, remember to take the HIGHEST power of each prime factor, not the lowest (that would give the GCF). Also, LCM(a, b) × GCF(a, b) = a × b is a useful check — if your answer does not satisfy this relationship, something went wrong.
Key Takeaways
- LCM is the smallest positive number that is divisible by all given numbers.
- Prime factorization method: take the highest power of each prime that appears in any number.
- Relationship: LCM(a,b) × GCF(a,b) = a × b. Use this to verify your answers.
- If two numbers are coprime (GCF = 1), then LCM = their product.
- LCM is used to find the LCD (Least Common Denominator) when adding/subtracting fractions.
- Real-world applications include scheduling, synchronization, and repeating cycle problems.