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Modular Arithmetic Calculator

Reduce an integer into its least nonnegative residue modulo a positive integer. This is useful for cyclic numbering, clock-style arithmetic and congruence exercises. A negative dividend still returns a residue between zero and m−1 rather than a negative remainder.

Enter every field in the stated units. Decimal and scientific notation are accepted for non-integer fields; blanks, commas, fractions and non-finite entries are rejected. Zero is accepted only where a field permits it.

Least nonnegative residue3
Euclidean quotient-4

How to use this calculator

  1. Identify the quantities required by the model and use the field units.
  2. Replace the example values, then select Calculate or press Enter.
  3. Read the labeled outputs and check the interpretation below.

Formula and interpretation

Residue r is the unique integer 0≤r<m with a=qm+r; quotient q=(a−r)/m.

Calculations run in your browser. The previous results are hidden when an input changes or validation fails. This prevents an old answer from being mistaken for the new calculation. Results are displayed to approximately ten significant digits using floating-point arithmetic; that precision does not imply equal precision in measured inputs.

Worked example

Integer dividend a = -17; Positive integer modulus m = 5. Least nonnegative residue = 3; Euclidean quotient = -4.

Substitute the example inputs into the stated formula in the same order as the labeled fields. These are the default values shown above, so the explanation and the working interface describe the same case.

Limitations

Inputs are exact integers in the supported ±1,000,000 range. This tool does not implement modular inverses, arbitrary-precision cryptography or floating-point remainders.

This is an educational mathematical model. Check the assumptions and units for your situation. Geometry results do not establish material strength, and probability outputs do not predict an individual outcome.

Frequently asked questions

Why is −17 mod 5 equal to 3?

Because −17 = (−4)×5+3, and 3 lies in the required residue range.

Why does the calculator reject my entry?

Each field shows its supported range. Integer fields reject fractions; mathematical conditions such as a zero divisor or invalid shape produce a separate explanation. Correct the indicated field or condition and calculate again.

How many digits should I use?

Use enough digits to retain your input accuracy. Round the final answer to the precision justified by the data; tiny residuals around theoretical zero can result from floating-point arithmetic.

Review notes and reference

Reviewed October 3, 2026. The calculator script was tested against the default example and separate boundary cases. Checks covered blank, text, comma-formatted, non-finite and out-of-range entries, plus recovery after errors. Background reference for the mathematical method. Explanations and worked examples are original.

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