LCM Calculator
Calculate the Least Common Multiple of multiple numbers with full prime factorization steps. Also computes GCF and shows how to find the LCD for fractions.
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How It Works
Factor each number into prime factors. Write in exponential form. Take the highest power of each prime that appears. Multiply together — that product is the LCM.
LCM = product of highest prime powersLCM(12,18): 12=2²×3, 18=2×3² → LCM=2²×3²=36LCM(a,b) = (a × b) / GCF(a,b). Find GCF first using the Euclidean algorithm, then divide the product of the numbers by the GCF.
LCM(a,b) = a × b / GCF(a,b)LCM(12,18) = 12×18/GCF(12,18) = 216/6 = 36To add 1/12 + 1/18, the LCD = LCM(12,18) = 36. Convert: 3/36 + 2/36 = 5/36. LCM gives the smallest common denominator, producing the simplest result.
LCD = LCM(all denominators)1/12 + 1/18 → LCD=36 → 3/36 + 2/36 = 5/36Two events repeat every a and b days. They coincide again after LCM(a,b) days. Example: events repeating every 6 and 9 days next coincide on day 18.
Next overlap = LCM(a, b)Every 6 and 9 days → LCM=18 → day 18Apply iteratively: LCM(a,b,c) = LCM(LCM(a,b), c). For 3, 4, 5: LCM(3,4)=12, then LCM(12,5)=60.
LCM(a,b,c) = LCM(LCM(a,b),c)LCM(3,4,5)=LCM(12,5)=60List multiples of each number until you find the smallest common one. Example: multiples of 4: 4,8,12,16… and 6: 6,12,18… First common is 12, so LCM(4,6)=12.
List multiples, find first matchLCM(4,6): 4,8,12 and 6,12 → 12Quick Reference
Common examples — verify instantly above.
Two numbers
LCM(4, 6)
12
Two numbers
LCM(12, 18)
36
Three numbers
LCM(3, 4, 5)
60
Coprime
LCM(7, 9)
63
LCD
1/6 + 1/4 LCD
12
Scheduling
Every 6 & 9 days
Day 18
Equal
LCM(12, 12)
12
One divides
LCM(6, 18)
18
Tips & Shortcuts
If one number divides the other evenly, the LCM is the larger number. LCM(6,18)=18 because 6 divides 18.
The LCD (Least Common Denominator) for adding fractions is simply the LCM of all denominators.
LCM(a,b) × GCF(a,b) = a × b. Use this to find LCM quickly: LCM = a×b / GCF.
For three or more numbers, apply LCM pairwise: LCM(a,b,c) = LCM(LCM(a,b), c).
Coprime numbers (GCF=1) have LCM = product. LCM(8,9) = 72 because GCF(8,9)=1.
The listing multiples method is easiest for small numbers but impractical above 100 — use prime factorization instead.
Common Mistakes
Confusing LCM and GCF — using GCF for adding fractions
Use LCM to find common denominators (LCD). Use GCF to simplify fractions. They serve opposite purposes.
Thinking LCM(a,b) = a × b always
Only true when GCF=1. LCM(12,18) = 36, not 216. The formula is LCM = a×b/GCF.
Stopping at the first common multiple found in a list
The first common multiple in a list is the LCM. If you find 12 for LCM(4,6), that is correct — do not continue looking.
Taking highest prime powers for GCF instead of LCM
GCF uses lowest powers of common primes. LCM uses highest powers of all primes. Opposite rules.
Using LCM formula for 3 numbers as LCM = a×b×c / GCF
For 3 numbers, apply iteratively: LCM(a,b,c) = LCM(LCM(a,b), c). There is no simple single-formula shortcut.
Expecting LCM to be smaller than the inputs
LCM is always at least as large as the largest input. If you get a smaller result, recalculate.
Frequently Asked Questions
The LCM (Least Common Multiple) is the smallest positive integer divisible by all given numbers. LCM(4,6)=12 because 12 is the smallest number that both 4 and 6 divide into evenly.
Use prime factorization: factor each number, take the highest power of each prime factor, multiply together. Or use LCM(a,b) = a×b/GCF(a,b).
Adding fractions with different denominators (LCD = LCM), scheduling repeating events, finding the first time two cycles coincide.
Yes — LCM is always at least as large as the largest input number. LCM equals the largest input only if it is a multiple of all others.
If GCF(a,b)=1, then LCM(a,b) = a×b. For example LCM(8,9)=72 because GCF(8,9)=1.
Find LCM(a,b), then find LCM(result, c), and so on. Order does not matter.
Find LCM of the first two numbers, then LCM of that result with the third, and so on. Example: LCM(4, 6, 10). LCM(4,6) = 12. LCM(12,10) = 60. Verify: 60÷4=15 ✓, 60÷6=10 ✓, 60÷10=6 ✓. The prime factorization shortcut works directly for multiple numbers: list all primes, take the highest power of each prime appearing in any of the numbers.
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