NUMBER THEORY / COMBINED RESIDUE SYSTEM
Combine two congruences with coprime moduli into one least nonnegative solution.
METHOD / WORKED EXAMPLE
The two-congruence theorem combines separate repeating residue conditions into one repeating solution class when the moduli are coprime. Both returned residues remain visible as the decisive check.
WORKED DEFAULT
For x congruent to 2 modulo 3 and x congruent to 3 modulo 5, the least solution is 8. Every solution is 8 plus a multiple of 15.
READ THE RESULT
The least result is one representative of an infinite class modulo the product. A failure for shared-factor moduli does not prove inconsistency; it means this coprime-only workflow is insufficient.
ASSUMPTIONS AND LIMITS
This page handles two coprime moduli only. Compatible non-coprime systems require a generalized CRT solver with additional consistency checks.
COMMON QUESTIONS
The two coprime residue patterns repeat together after the product of their moduli, here 3 x 5 = 15. Therefore 8, 23, 38, and every value 8 + 15k satisfy the same pair of congruences. Reporting the least nonnegative representative gives a canonical answer without discarding the full solution class.
A solution may still exist if the residue difference is divisible by the moduli's greatest common divisor, but the combined period becomes their least common multiple rather than their product. This page intentionally rejects that case instead of silently applying the coprime formula. Use a generalized CRT method with an explicit compatibility check.
Substitute the returned solution into each original modulus. Its remainder must equal the corresponding normalized input residue. Both checks are necessary: satisfying only one congruence does not solve the system. The combined-modulus statement then describes every equivalent solution, while the displayed result is simply the least nonnegative member of that class.
RELATED TOOLS
Use boundary
Normalize both residues, invert the first modulus modulo the second, solve the required step count, and verify both residues in the combined modulus.
Combine x = a (mod m) and x = b (mod n); for coprime m,n the solution is unique modulo mn.