GCF Calculator
Find the Greatest Common Factor and Least Common Multiple of multiple numbers at once. Includes prime factorization, factor listing, and step-by-step work.
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How It Works
Factor each number into primes. The GCF is the product of the lowest power of each common prime factor. The LCM is the product of the highest power of each prime factor.
GCF: common primes at min powerGCF(24,36): 24=2³×3, 36=2²×3² → GCF=2²×3=12Repeatedly divide the larger number by the smaller and replace with the remainder until remainder is 0. The last non-zero remainder is the GCF. Much faster than factoring for large numbers.
GCF(a,b) = GCF(b, a mod b)GCF(48,18): 48=2×18+12 → GCF(18,12)=GCF(12,6)=6Once GCF is known, LCM = (a × b) / GCF. This formula works for two numbers and is the most efficient way to find LCM after finding GCF.
LCM(a,b) = a × b / GCF(a,b)LCM(12,18) = 12×18/6 = 216/6 = 36Apply the Euclidean algorithm iteratively: GCF(a,b,c) = GCF(GCF(a,b), c). Similarly for LCM: LCM(a,b,c) = LCM(LCM(a,b), c).
GCF(a,b,c) = GCF(GCF(a,b), c)GCF(12,18,24) = GCF(GCF(12,18),24) = GCF(6,24)=6When GCF = 1, numbers are called coprime or relatively prime. In this case, LCM = a × b. Consecutive integers are always coprime.
GCF(a,b)=1 → LCM=a×bGCF(8,9)=1, so LCM(8,9)=72To simplify 24/36, find GCF(24,36) = 12. Divide both numerator and denominator by 12: 24÷12=2, 36÷12=3. Result: 2/3. GCF always gives the fully simplified fraction in one step.
24/36 ÷ GCF(24,36)/GCF(24,36) = 2/3GCF(24,36)=12 → 24/36 = 2/3Quick Reference
Common examples — verify instantly above.
GCF
GCF(12, 18)
6
GCF
GCF(24, 36, 48)
12
LCM
LCM(4, 6)
12
LCM
LCM(12, 18)
36
Coprime
GCF(8, 9)
1
Coprime
LCM(8, 9)
72
Large
GCF(270, 192)
6
Fractions
24/36 simplified
2/3
Tips & Shortcuts
To simplify a fraction instantly: divide numerator and denominator by their GCF. GCF(numerator, denominator) gives the simplification factor in one step.
GCF(a,b) × LCM(a,b) = a × b — use this to find LCM quickly once you know GCF, or vice versa.
If one number divides the other evenly, the GCF is the smaller number and the LCM is the larger.
Two consecutive integers always have GCF = 1 (they are coprime). Example: GCF(7,8) = 1.
To add fractions 1/12 + 1/18: find LCM(12,18)=36, then 3/36 + 2/36 = 5/36. LCM gives the LCD.
The Euclidean algorithm is dramatically faster for large numbers. GCF(1234567, 890123) is solved in a few steps vs. thousands of trial divisions.
Common Mistakes
Confusing GCF and LCM — using LCM to simplify fractions
Use GCF (not LCM) to simplify fractions. Use LCM (not GCF) to find common denominators.
Thinking GCF of two primes is one of the primes
Two different primes always have GCF = 1. Example: GCF(7,11) = 1, not 7 or 11.
Using GCF formula LCM = a×b/GCF for 3+ numbers
For 3 numbers: find LCM(a,b) first, then LCM(result, c). Do not multiply all three and divide by one GCF.
Forgetting that LCM ≥ each input number
LCM is always at least as large as the largest input. If your result is smaller, recalculate.
Stopping prime factorization too early
Continue factoring until all factors are prime. 12 = 4×3 is not done — 4 = 2×2, so 12 = 2×2×3.
Calculating GCF by listing factors for large numbers
Listing all factors of 1,234,567 is impractical. Use the Euclidean algorithm for large numbers.
Frequently Asked Questions
The GCF (Greatest Common Factor) is the largest number that divides evenly into all given numbers. GCF(12,18) = 6 because 6 is the largest number that divides both 12 and 18.
The LCM (Least Common Multiple) is the smallest positive number that is a multiple of all given numbers. LCM(4,6) = 12 because 12 is the smallest number divisible by both 4 and 6.
For two numbers a and b: GCF(a,b) × LCM(a,b) = a × b. This relationship lets you find LCM from GCF quickly.
Two different prime numbers always have a GCF of 1, since primes have no factors other than 1 and themselves.
GCF is used to simplify fractions, split items into equal groups, and tile floors with the largest possible square tiles.
LCM is used to find when two repeating events coincide, add fractions with different denominators, and schedule recurring events.
GCF simplifies fractions: 12/18 ÷ GCF(12,18)=6 → 2/3. LCM finds common denominators: to add 1/4 + 1/6, use LCM(4,6)=12 → 3/12 + 2/12 = 5/12. In scheduling: two events repeating every 12 and 18 minutes next coincide after LCM(12,18)=36 minutes. The GCF answers "how big can each group be?" while LCM answers "when do cycles align?"
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