GCF and LCM Calculator – Greatest Common Factor & Least Common Multiple | CalcVelo
🔢 Math Tool · GCF + LCM Together
GCF and LCM Calculator
Find the Greatest Common Factor (GCF) and Least Common Multiple (LCM) of 2 or 3 numbers instantly — with step-by-step prime factorization and Euclidean algorithm breakdown.
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Enter Your Numbers
Whole positive numbers only (1 or greater)
Examples:
GCF / GCD / HCF
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Greatest Common Factor
LCM / LCD
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Least Common Multiple
Numbers—
GCF (Greatest Common Factor)—
LCM (Least Common Multiple)—
GCF × LCM—
🔗 GCF × LCM = A × B
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📐 GCF — Prime Factorization Method
📐 LCM — Prime Factorization Method
How to Find GCF and LCM
This free GCF and LCM Calculator finds both the Greatest Common Factor and Least Common Multiple simultaneously — saving you time when you need both values (which is most of the time in school math).
GCF Formula (Prime Factorization)
GCF = Product of COMMON prime factors with LOWEST exponents
Example: GCF(12,18) → 12=2²×3, 18=2×3² → GCF=2¹×3¹=6
LCM Formula (Prime Factorization)
LCM = Product of ALL prime factors with HIGHEST exponents
Example: LCM(12,18) → 12=2²×3, 18=2×3² → LCM=2²×3²=36
GCF × LCM Relationship
GCF(A,B) × LCM(A,B) = A × B
Example: GCF(12,18)=6 and LCM(12,18)=36 → 6×36=216=12×18 ✓
When to Use GCF vs LCM
GCF — use for: Simplifying fractions (divide numerator and denominator by GCF), splitting things into equal groups, finding common units
LCM — use for: Adding and subtracting fractions (find common denominator), scheduling repeating events, finding when two cycles sync
Real-World Examples
Simplify 12/18: GCF(12,18)=6 → divide both by 6 → 2/3 ✓
Add 1/4 + 1/6: LCM(4,6)=12 → 3/12 + 2/12 = 5/12 ✓
Event scheduling: Bus A every 12 min, Bus B every 18 min → LCM=36 min until both arrive together
Tile cutting: Tiles of 12cm and 18cm → GCF=6cm is the largest equal piece
GCF Also Known As
GCF (Greatest Common Factor) = GCD (Greatest Common Divisor) = HCF (Highest Common Factor). All three names refer to the same concept — the largest number that divides evenly into all given numbers.
LCM Also Known As
LCM (Least Common Multiple) = LCD (Least Common Denominator) = LCM (Lowest Common Multiple). All refer to the same concept — the smallest number that all given numbers divide into evenly.
Euclidean Algorithm for GCF
The Euclidean algorithm is a fast method for finding GCF. Divide the larger number by the smaller, take the remainder, then repeat until the remainder is 0. The last non-zero remainder is the GCF.
LCM(A, B) = (A × B) ÷ GCF(A, B)
Example: LCM(12, 18) = (12 × 18) ÷ 6 = 216 ÷ 6 = 36 ✓
GCF and LCM of 3 Numbers
For three numbers A, B, C: find GCF(A,B) first, then find GCF of that result with C. Same approach for LCM. This calculator handles 3-number GCF and LCM automatically.
GCF is the largest number that divides evenly into all given numbers. GCF(12,8)=4 because 4 is the largest number dividing both. Also called GCD or HCF. Used to simplify fractions — divide numerator and denominator by GCF.
LCM is the smallest positive number divisible by all given numbers. LCM(4,6)=12 because 12 is the smallest number both 4 and 6 divide into evenly. Used to find common denominators when adding or subtracting fractions.
Method 1 (Prime Factorization): Factor each number → find common prime factors → multiply common factors with lowest exponents. Example: GCF(12,18): 12=2²×3, 18=2×3² → common: 2¹×3¹=6. Method 2 (Euclidean): Divide larger by smaller, take remainder, repeat until remainder=0 — divisor is GCF.
For any two numbers A and B: GCF(A,B) × LCM(A,B) = A × B. So if you know the GCF you can find the LCM: LCM = (A×B) ÷ GCF. Example: GCF(4,6)=2 → LCM=(4×6)÷2=12. Verify: 2×12=24=4×6 ✓
GCF is used to simplify fractions, split items into equal groups, find largest tile/piece size that fits evenly, and reduce ratios. Example: Simplify 18/24 — GCF(18,24)=6 → 18÷6=3, 24÷6=4 → simplified fraction is 3/4.
LCM is used to add/subtract fractions (find common denominator), schedule repeating events (when do two cycles coincide?), and find common periods. Example: Add 1/4+1/6 — LCM(4,6)=12 → 3/12+2/12=5/12.
GCF(12, 18) = 6. Prime factorization: 12 = 2² × 3 and 18 = 2 × 3². Common factors with lowest exponent: 2¹ × 3¹ = 6. Verify: 12÷6=2 and 18÷6=3 — both divide evenly. Also, LCM(12,18) = 36 and 6×36 = 216 = 12×18 ✓.
LCM(4, 6) = 12. Prime factorization: 4 = 2² and 6 = 2 × 3. Take highest powers: 2² × 3 = 12. Multiples of 4: 4,8,12,16... Multiples of 6: 6,12,18... First common = 12. LCM is used to add 1/4 + 1/6 = 3/12 + 2/12 = 5/12.
CalcVelo GCF and LCM Calculator — Free online tool to find Greatest Common Factor (GCF/GCD/HCF) and Least Common Multiple (LCM/LCD) with step-by-step prime factorization and Euclidean algorithm solutions.
