Remainder Calculator

This professional calculator finds the integer quotient and remainder for any integer division, including negative numbers. Choose among Euclidean (non-negative remainder), floored (Python semantics), and truncated (C/Java semantics) to match your math work or programming language.

Interactive Calculator

Division mode *
Choose how quotient and remainder are defined, especially for negative numbers.
Enter an integer value for the dividend a.
Enter a nonzero integer for the divisor d.

Results

Mode—
Quotient (q)—
Remainder (r)—
Identity check—
Notes—

For integers beyond ~9×1015, JavaScript may lose precision; a warning will appear.

Source & Methodology

Authoritative source: Euclidean (division with remainder) theorem — see Euclidean division (encyclopedic reference) and standard number-theory texts. All computations follow the identity \(a = q\cdot d + r\) with \(q,r\in\mathbb{Z}\) and mode-specific constraints on \(r\).
Tutti i calcoli si basano rigorosamente sulle formule e sui dati forniti da questa fonte. (All calculations strictly follow the referenced formulas.)

The Formula Explained

For integers \(a\) (dividend) and \(d\ne 0\) (divisor), find integers \(q\) and \(r\) such that

$$ a = q\,d + r $$

Under Euclidean convention: $$ 0 \le r < |d|. $$

Floored (Python): \( q = \left\lfloor \dfrac{a}{d} \right\rfloor,\quad r = a - qd.\)

Truncated (C/Java): \( q = \mathrm{trunc}\!\left(\dfrac{a}{d}\right),\quad r = a - qd.\)

Glossary of Variables

  • Dividend (a): Integer being divided.
  • Divisor (d): Integer you divide by (nonzero).
  • Quotient (q): Integer result of the division, per chosen mode.
  • Remainder (r): Integer “left over.” Constraints depend on mode.
  • Identity check: Verifies \(a = q\cdot d + r\).

How It Works: A Step-by-Step Example

Goal: Divide \(a=-23\) by \(d=5\) using Euclidean remainder.

  1. Compute \(q=\left\lfloor \frac{-23}{5} \right\rfloor=\lfloor-4.6\rfloor=-5\).
  2. Compute \(r=a-qd=-23-(-5\cdot5)=2\).
  3. Check \(a=qd+r\): \(-23=(-5)\cdot5+2=-25+2=-23\) ✓ and \(0\le2<|5|\).

FAQ

Which mode should I choose?

Pick Euclidean for number theory and modular arithmetic; Floored to match Python; Truncated to match C/Java behavior.

Can the remainder be negative?

Yes under truncated division when signs differ. Euclidean guarantees \(0\le r<|d|\).

Does this handle negative inputs?

Yes; the calculator applies the correct formulas for each mode and shows the identity \(a=qd+r\).

Why do different languages give different answers?

They define \(q\) differently: floor vs truncation. See the references above for Python and C nuances.

What about very large integers?

JavaScript is exact for integers up to 253−1 (~9×1015). Beyond that, results may lose precision; the UI warns you.

Is dividing by zero allowed?

No. Division by zero is undefined and the tool prevents it.

Strumento sviluppato da Ugo Candido. Contenuti verificati da CalcDomain Editorial Team.
Ultima revisione per l'accuratezza in data: .