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Chinese Remainder Theorem — result sheet
Solve two simultaneous congruences with coprime positive moduli.
Inputs used
Results
Visual chart
Breakdown
Calculation steps
Returned data table
Formula and methodology
Formula: For coprime m1 and m2, solve x = r1 (mod m1) and x = r2 (mod m2) with one solution modulo m1 x m2.
The calculator normalizes both remainders, combines the congruences with a modular inverse, and returns the smallest nonnegative solution plus the combined modulus.
This result follows the calculator's declared inputs, precision, validation boundaries, and model limits.
Input contract
- First remainder — minimum -1000000000; maximum 1000000000
- First modulus — minimum 2; maximum 1000000
- Second remainder — minimum -1000000000; maximum 1000000000
- Second modulus — minimum 2; maximum 1000000
Worked example
| Input | Value |
|---|---|
| First remainder | 2 |
| First modulus | 3 |
| Second remainder | 3 |
| Second modulus | 5 |
x = 8 modulo 15.
Assumptions and limits
- Both moduli are positive integers at least 2.
- The two moduli are coprime, so the simple two-congruence CRT contract applies.
- The returned solution is the least nonnegative representative of its congruence class.
Calculator note
Source and methodology
Use the official WorldCalculate methodology policy for the source, formula, precision, and boundary standards behind this calculator.
Planning estimate, not financial, medical, legal, or professional advice. © WorldCalculate — reuse with attribution. Built and curated by Hassan ALRowaie.
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