Congruence Calculator

Practical meaning

Congruences model repeating schedules, divisibility constraints, cyclic positions, and modular equations. A related application of Congruence is remainder convention.

Calculating it by hand

Divide a, b, and n by their common divisor, invert the reduced coefficient, find one residue, then add steps of the reduced modulus. To continue from Congruence, try coefficient inverse.

What to record with the answer

Before entering coefficient a, target b and positive modulus, identify what each field represents. Its label determines how the congruence relationship interprets the entry. The browser cannot supply missing unit context, so preserve that context in your notes.

Adding n to any solution produces the same residue class. That behavior gives the congruence output a built-in reasonableness test.

A final answer should carry enough context to be reusable: name it solution residues, attach any applicable unit, and record the rounding point. If another person must reproduce the result, the supplied inputs in the fields and the displayed formula are more informative than the decimal alone.

A modular inverse solves the special coprime step; this page also handles multiple or absent solutions. Writing “Solution residues” beside the output prevents that mix-up.

A useful note for this result contains the supplied inputs (coefficient a, target b and positive modulus), the answer label, and the chosen precision. That record can be checked without recreating the entire page state.

Why several solutions appear

When g=gcd(a,n) divides b, reducing by g creates one solution modulo n/g. Viewed in the original modulus n, that solution repeats g times at intervals of n/g. Listing all standard residues prevents a correct but incomplete single answer from hiding the other classes.