How to Calculate GCD and LCM
Our GCD/LCM calculator finds the Greatest Common Divisor (GCD) and Least Common Multiple (LCM) of two or more numbers. Essential for simplifying fractions, finding common denominators, and solving number theory problems.
Understanding GCD and LCM
The GCD (also called GCF or HCF) is the largest number that divides all given numbers evenly. The LCM is the smallest number that all given numbers divide into evenly. These concepts are fundamental to working with fractions and ratios.
GCD and LCM Formulas
GCD uses the Euclidean algorithm: repeatedly divide larger by smaller until remainder is 0. LCM = (a × b) / GCD(a, b). For multiple numbers, apply iteratively.
Example:
GCD(48, 18): 48 = 18×2 + 12, 18 = 12×1 + 6, 12 = 6×2 + 0 → GCD = 6. LCM(48, 18) = (48 × 18) / 6 = 144.
Common Use Cases
Real-world applications for this calculator
Simplifying Fractions
Divide numerator and denominator by GCD to reduce to lowest terms.
Adding Fractions
LCM of denominators gives the least common denominator.
Scheduling Problems
LCM helps find when periodic events coincide.
Tips
- GCD × LCM = product of the two numbers.
- GCD helps reduce fractions; LCM helps add fractions.
- Coprime numbers (GCD = 1) have LCM equal to their product.
- Prime factorization also works: GCD uses lowest powers, LCM uses highest.
Frequently Asked Questions
What is the difference between GCD and LCM?
GCD is the largest number that divides all given numbers evenly (used to simplify fractions). LCM is the smallest number that all given numbers divide into (used to find common denominators).
How do I find the GCD of two numbers?
Use the Euclidean algorithm: divide the larger by the smaller, then divide the divisor by the remainder. Repeat until remainder is 0. The last non-zero remainder is the GCD.
How do I calculate LCM?
Once you have the GCD, use LCM(a, b) = (a × b) / GCD(a, b). Or list multiples of each number until you find the smallest common one.
What does it mean if GCD is 1?
If GCD(a, b) = 1, the numbers are "coprime" or "relatively prime"-they share no common factors other than 1. Examples: 8 and 15, 7 and 9.
How do I find GCD/LCM of more than two numbers?
Apply the operation iteratively: GCD(a, b, c) = GCD(GCD(a, b), c). Same for LCM: LCM(a, b, c) = LCM(LCM(a, b), c).