GCD and LCM Calculator

Two whole numbers nearly always share a hidden skeleton: the biggest block that fits in both, and the smallest multiple that both build together. The GCD and the LCM answer exactly those two questions. They surface in fraction reduction, common denominators, gear ratios, scheduling and modular arithmetic - and both collapse into a handful of exact steps once the division identity is known.

GCD and LCM Calculator takes any two integers, positive or negative, and runs the Euclidean algorithm with BigInt precision. In one breath it reports the greatest common divisor, the least common multiple, and the Bezout identity ax + by = gcd with the exact coefficients x and y, verified on every run. Everything happens locally in the browser.

• BigInt exactness: Fifty-digit inputs with zero rounding - gcd, lcm and coefficients all precise.
• GCD and LCM together: Both linked by the product identity, returned on every run.
• Bezout identity: The ax + by = gcd equation with exact integer coefficients, verified each time.
• Coprime flag: gcd 1 announces the pair is relatively prime instantly.
• Product link shown: gcd x lcm equals a x b - verified by hand every run.
• Step counter: The number of Euclidean steps is reported on the run.
• Copy the result: One click puts the full triplet on the clipboard.
• Free and private: Nothing is uploaded or stored; the math runs in your tab.

1. Enter a and b - any integers, negative signs welcome, any length.
2. Calculate - the Euclidean algorithm runs exactly on both values.
3. Read the triplet - gcd, lcm, and the verified Bezout identity.
4. Spot the coprimality - a gcd of 1 is flagged immediately.
5. Copy the full result row for your records.

Example 1 - Classic pair. a = 12, b = 18: the gcd is 6, the lcm is 36, and Bezout gives 12( -1 ) + 18( 1 ) = 6.

Example 2 - Smallest multiple. a = 4, b = 6: gcd 2, lcm 12, and 2 = 4( 1 ) + 6( -1 ) - verified and displayed.

Example 3 - Coprime pair. a = 17, b = 5: gcd 1 (relatively prime), lcm 85, and coefficients build 1 exactly: 17( -2 ) + 5( 7 ) = 1.

Example 4 - Edge cases. a = 0, b = 7: gcd 7, lcm 0 - handled cleanly by the Euclidean setup; and a = 0, b = 0 is flagged as undefined.

Example 5 - Huge inputs. a and b with fifty digits each: a few Euclidean steps collapse them to their exact gcd, with the Bezout identity verified at full precision.

• One run, three answers: GCD, LCM and Bezout together in a breath.
• Verified mathematics: The ax + by = gcd identity is checked per run.
• Any size, exact: BigInt keeps hundreds of digits lossless.
• Coprime detection: gcd 1 flags relative primality instantly.
• Edge-case safe: Zeros and negatives handled cleanly, no crashes.
• Private and free: All computation is local; nothing is logged.