Chinese Remainder Theorem Calculator

Background.

Chinese Remainder Theorem Calculator evaluates least nonnegative solution of two congruences with positive coprime integer moduli. The page keeps every model input visible and uses the relationship x ≡ a (mod m); x ≡ b (mod n), gcd(m,n)=1. It is designed for a transparent calculation where the quantities have already been measured or selected from an appropriate source. It does not choose a material, operating condition, reference state, or empirical coefficient on the user's behalf.

Enter first remainder a, first modulus m, second remainder b, second modulus n in the units printed beside the fields. These are coherent SI quantities, so the displayed equation can be followed without a hidden unit factor. A result is only comparable with another source when the same quantity definitions, reference conditions, and sign or magnitude convention are used. Record those conditions whenever the number supports engineering, laboratory, or coursework decisions.

The calculator performs arithmetic with Decimal.js and rounds once at the output boundary to twelve significant digits. That protects very small and very large scientific results from early decimal-place rounding. The tests do more than pin one example: they check the dimensional scaling implied by each variable, finite and positive domain guards, several orders of magnitude, and the formula-engine registration used by the live page.

Two-congruence coprime-modulus case with canonical remainders; it does not parse symbolic systems, handle noncoprime compatible systems, or accept noninteger inputs. The scope statement appears beside the numerical result because it changes how the answer may be used. A neat number does not remove uncertainty in measurements, material properties, geometry, calibration, or the assumptions used to reduce a real system to one equation.

Use scaling as a quick reasonableness check. If an input appears in the numerator, increasing it should move the result in the same direction; a denominator should move it in the opposite direction; a square-root term changes more slowly. If the page behaves differently from the displayed relationship, stop and review the units. The calculator rejects zero, negative, infinite, and nonnumeric quantities where the equation requires a positive magnitude.

What is chinese remainder theorem calculator?

Chinese Remainder Theorem Calculator is a transparent implementation of x ≡ a (mod m); x ≡ b (mod n), gcd(m,n)=1 for least nonnegative solution of two congruences with positive coprime integer moduli.

How to use this calculator.

1. 01Confirm that the displayed quantity equation matches the model you intend to use.
2. 02Convert every measurement to the SI unit printed beside its field.
3. 03Enter sourced magnitudes and keep their reference conditions with the result.
4. 04Read the numeric result together with the model-scope output.
5. 05Round the reported value to the uncertainty supported by the inputs.

The formula.

The implementation evaluates x ≡ a (mod m); x ≡ b (mod n), gcd(m,n)=1 with Decimal.js. Inputs are required to be finite and positive because this page treats them as magnitudes. Arithmetic is not rounded between operations; each numeric output is rounded once to twelve significant digits. The scaling tests independently verify the power of every input in the equation.

A worked example.

Frequently asked questions.

References& sources.

1. [1]NIST Digital Library of Mathematical Functions, section 27.14, Chinese Remainder Theorem.
2. [2]OpenStax, Contemporary Mathematics, modular arithmetic.
3. [3]BIPM, The International System of Units (SI Brochure), 9th ed., version 3.01, coherent derived units and quantity equations.