LCM and GCD Calculator

What GCD and LCM Are

Greatest Common Divisor (GCD), also called Highest Common Factor (HCF): the largest number that divides all the given numbers exactly (without remainder). Example: GCD of 12 and 18. Factors of 12: 1, 2, 3, 4, 6, 12. Factors of 18: 1, 2, 3, 6, 9, 18. Common factors: 1, 2, 3, 6. Greatest: 6. So GCD(12, 18) = 6. Lowest Common Multiple (LCM): the smallest number that all the given numbers divide into exactly. Example: LCM of 12 and 18. Multiples of 12: 12, 24, 36, 48, 60... Multiples of 18: 18, 36,

Finding GCD

Method 1 — Prime factorisation: break each number into prime factors. 12 = 2² × 3. 18 = 2 × 3². GCD = product of LOWEST power of each common prime. Common primes: 2 (lowest power 2¹) and 3 (lowest power 3¹). GCD = 2 × 3 = 6. Method 2 — Euclidean algorithm (faster for large numbers): repeatedly divide and take remainders. GCD(48, 18): 48 = 2×18 + 12. GCD(18, 12): 18 = 1×12 + 6. GCD(12, 6): 12 = 2×6 + 0. Remainder is 0, so GCD = 6 (the last non-zero remainder). The Euclidean algorithm is extremely

Finding LCM

Method 1 — Prime factorisation: break each into primes. 12 = 2² × 3. 18 = 2 × 3². LCM = product of HIGHEST power of each prime appearing. Primes: 2 (highest 2²) and 3 (highest 3²). LCM = 2² × 3² = 4 × 9 = 36. Method 2 — Using GCD: LCM(a, b) = (a × b) / GCD(a, b). LCM(12, 18) = (12 × 18) / 6 = 216 / 6 = 36. Fast once you have GCD. For more than two numbers: LCM(a, b, c) = LCM(LCM(a, b), c). Apply pairwise. Note: the GCD × LCM = product relationship only works for TWO numbers, not three or more. F

Real-World Applications

Fractions: GCD simplifies fractions to lowest terms. 12/18 = (12÷6)/(18÷6) = 2/3. LCM finds common denominators. 1/12 + 1/18: LCM = 36. = 3/36 + 2/36 = 5/36. Scheduling/timing: two buses leave at intervals of 12 and 18 minutes. When do they coincide? LCM(12,18) = 36 minutes. Three machines cycle every 4, 6, 8 seconds — synchronise every LCM(4,6,8) = 24 seconds. Tiling and packing: largest square tile fitting a 12×18 area without cutting: GCD(12,18) = 6cm tiles. Gear ratios: teeth counts and rota