Extended Euclidean Calculator

How the extended euclidean calculator works

The ordinary Euclidean algorithm finds a greatest common divisor; the extended version also expresses that divisor as an integer combination of the two inputs. While remainders shrink, this calculator updates two coefficient tracks alongside them. The final coefficients x and y satisfy a times x plus b times y equals the gcd, creating the Bézout identity needed for modular inverses and linear Diophantine equations.

For the default inputs 240 and 46, repeated division reaches a greatest common divisor of 2. Back-substitution is represented by the tracked coefficients minus 9 and 47. The verified identity is 240 times minus 9 plus 46 times 47 equals 2. The output prints that arithmetic check so coefficient signs can be verified immediately.

Reading extended euclidean calculator results privately

Bézout coefficients are not unique: adding suitable multiples of the reduced input pair creates other valid solutions. The calculator reports the particular pair produced by its quotient sequence. Both inputs may be negative, but they cannot both be zero because that gcd is undefined here. Safe-integer limits protect the intermediate products from silent precision loss. This extended euclidean calculator calculation runs entirely in your browser, so the numbers you enter never leave your device.