Congruência Linear

The linear congruence calculator solves equations in the form ax ≡ b (mod m), where a, b, and m are integers. This type of equation is fundamental in number theory and cryptography. To find solutions, the calculator checks if the greatest common divisor (gcd) of a and m divides b. If so, the solution x is computed using the modular multiplicative inverse of a, adjusted according to the modulus.

It works by applying modular arithmetic techniques. First, it calculates gcd(a, m) using the Euclidean algorithm. If gcd(a, m) does not divide b, no solution exists. Otherwise, the equation is reduced using Bézout's identity, finding x = (b * inverse(a)) mod m. Solutions can be multiple, depending on the modulus and gcd.

Use this calculator for problems involving cycles, scheduling, or cryptographic systems. For instance, in periodic task programming or RSA key analysis. Be cautious with non-coprime a and m values: this can limit or invalidate solutions. Always simplify the equation before using the calculator.

It is essential to understand that linear congruence requires a and m to be positive integers. If gcd(a, m) does not divide b, the equation has no solution. The final result is x ≡ c (mod n), where n is m divided by gcd(a, m). Fractional values must be converted to integers before being entered into the calculator.