LCM and GCD Calculator
Find the Lowest Common Multiple (LCM) and Greatest Common Divisor / Highest Common Factor (GCD/HCF) of any two or three numbers, with full working shown.
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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, 54... The first common multiple appearing in both lists, 36, is the LCM.
What should I know about 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 efficient even for very large numbers, since it converges in relatively few steps regardless of how big the starting values are.
What do I need to know about 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. For three or more numbers, calculate the LCM of the first two, then find the LCM of that result with the next number, repeating until all numbers are included.
What's the key thing to understand about 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 rotation speeds of meshing gears use LCM to determine how many rotations each gear completes before the same teeth mesh together again.