Chinese Remainder Theorem Calculator

About this calculator

The Chinese Remainder Theorem says that if you know the remainder of an unknown number x when divided by several moduli that are pairwise coprime (no two of them share a common factor), then there is exactly one value of x, modulo the product of all those moduli, that satisfies every congruence x ≡ r (mod m) simultaneously. The theorem takes its name from an old Chinese mathematics text that posed the puzzle of counting a group of soldiers using their remainders when lined up in rows of different sizes.

Beyond the classic puzzle, CRT is genuinely load-bearing in modern computing: RSA implementations use it to speed up private-key decryption by working modulo each prime factor separately and recombining the results, and it underlies calendar arithmetic, certain error-detecting codes, and hashing schemes that split a large modulus into smaller coprime pieces.

This calculator takes two or three congruences x ≡ r (mod m), checks that the moduli are pairwise coprime, and returns the unique solution modulo their product — along with the working so you can see how it was combined.