Chinese Remainder Theorem Calculator
Solve systems of congruences using the Chinese Remainder Theorem. Find x such that x ≡ r1 (mod m1) and x ≡ r2 (mod m2).
Inputs
Results
Solution (x)
8
Combined Modulus15
Verification8 mod 3 = 2, 8 mod 5 = 3
How to Use This Calculator
- Enter Remainder 1 (r₁), Modulus 1 (m₁), and Remainder 2 (r₂).
- Set Modulus 2 (m₂).
- Review the Solution (x) result.
- Use Combined Modulus and Verification to inform your decision.
- Use the chart to visualize the results and explore different scenarios by adjusting inputs.
What each input means
- Remainder 1 (r₁)
- First remainder: x ≡ r₁ (mod m₁)
- Modulus 1 (m₁)
- First modulus
- Remainder 2 (r₂)
- Second remainder: x ≡ r₂ (mod m₂)
- Modulus 2 (m₂)
- Second modulus
How this is calculated
Worked example, using the default values
- Identify Input Parameters4 parametersRemainder 1 (r₁) = 2, Modulus 1 (m₁) = 3, Remainder 2 (r₂) = 3, Modulus 2 (m₂) = 5 = 4 input(s) provided
- Calculate Solution8 = 8
- Calculate Combined Modulus15 = 15
- Calculate VerificationVerification8 mod 3 = 2, 8 mod 5 = 3 = 8 mod 3 = 2, 8 mod 5 = 3
Engine last updated . Checked against 3 independently-derived tests — how we verify calculators.
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