What Are GCD and LCM?
The Greatest Common Divisor (GCD) is the largest positive integer that divides both numbers without a remainder. The Least Common Multiple (LCM) is the smallest positive integer that is divisible by both numbers. These two concepts are fundamental in mathematics and have practical applications in everyday problems.
How to Calculate
The Euclidean algorithm finds GCD efficiently by repeatedly applying GCD(a,b) = GCD(b, a mod b) until the remainder becomes 0. For example, GCD(48, 36): 48 mod 36 = 12 → GCD(36, 12): 36 mod 12 = 0 → GCD = 12. Once GCD is known, LCM = a × b ÷ GCD(a,b). For 48 and 36: LCM = 48 × 36 ÷ 12 = 144.
Real-World Uses
• Simplifying fractions: 24/36 → GCD(24,36)=12 → reduces to 2/3
• Adding fractions: 1/4 + 1/6 → LCM(4,6)=12 → 3/12 + 2/12 = 5/12
• Scheduling: two buses running every 12 and 18 minutes → LCM(12,18)=36 minutes between simultaneous departures
Frequently Asked Questions
Yes, for any two positive integers a and b, GCD(a,b) × LCM(a,b) = a × b always holds. This calculator uses this relationship to compute LCM efficiently.
Coprime numbers share no common factors other than 1, so their GCD is 1 and their LCM equals their product. For example, GCD(7, 9) = 1 and LCM(7, 9) = 63.