GCD & LCM Calculator
Find the Greatest Common Divisor and Least Common Multiple of two or more numbers with step-by-step working.
GCD (HCF)
6
Greatest Common Divisor
LCM
144
Least Common Multiple
Prime Factorization
48 = 2^4 × 3
18 = 2 × 3^2
LCM × GCD = 48 × 18 = 864 ✓
About the GCD & LCM Calculator
The Greatest Common Divisor (GCD) and Least Common Multiple (LCM) are foundational number theory concepts with surprisingly broad practical applications: simplifying fractions (use GCD), synchronizing repeating events (use LCM), and optimizing packaging or cutting problems (use GCD). The Euclidean algorithm finds GCD efficiently for any two numbers.
GCD and LCM
GCD(a,b): Euclidean algorithm - repeatedly replace (a,b) with (b, a mod b) until b=0 · LCM(a,b) = (a × b) / GCD(a,b)
GCD(48, 36): 48 mod 36 = 12, 36 mod 12 = 0 → GCD = 12 · LCM(48, 36) = (48×36)/12 = 144 · GCD = 1 means the numbers are coprime (no common factors)
Worked Example
Two buses leave a stop at the same time. Bus A every 15 min, Bus B every 20 min. When do they next leave together?
LCM(15, 20) = 60 minutes · Both buses next depart together after 60 minutes
Tips & Insights
- 1
GCD is used to reduce fractions to lowest terms - divide both numerator and denominator by GCD(numerator, denominator).
- 2
LCM is used to find the common denominator when adding or subtracting fractions.
- 3
In manufacturing, LCM determines when two production cycles next coincide - useful for maintenance scheduling.
Why this matters for you
GCD and LCM solve real scheduling, packing, and synchronization problems. Beyond textbook exercises, they appear in computer science (memory alignment), music theory (rhythmic patterns), and event planning (when multiple recurring events next coincide). The Euclidean algorithm for GCD is one of the oldest algorithms in mathematics and still runs efficiently on modern computers.
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