Input
Result
Enter values and press Calculate to see the result here.
Example Data
Use these examples to understand sign conventions and Euclidean vs truncated remainders.
| Dividend (a) | Divisor (b) | Euclidean (q, r) | Truncated (q, r) |
|---|---|---|---|
| -13 | 5 | (-3, 2) | (-2, -3) |
| 20 | 6 | (3, 2) | (3, 2) |
| 7 | -3 | (-2, 1) | (-2, 1) |
| 123 | 10 | (12, 3) | (12, 3) |
Comparison: Euclidean vs Truncated (More Samples)
Additional pairs showing how remainders differ with signs and decimals.
| a | b | qeuclid | reuclid | qtrunc | rtrunc | a/b (exact, 4dp) |
|---|---|---|---|---|---|---|
| -25 | 4 | -7 | 3 | -6 | -1 | -6.2500 |
| 25 | -4 | -6 | 1 | -6 | 1 | -6.2500 |
| -25 | -4 | 7 | 3 | 6 | -1 | 6.2500 |
| 10.5 | 3 | 3 | 1.5 | 3 | 1.5 | 3.5000 |
| -10.5 | 3 | -4 | 1.5 | -3 | -1.5 | -3.5000 |
| 7.2 | -2.5 | -2 | 2.2 | -2 | 2.2 | -2.8800 |
Language Modulo Behavior (Quick Reference)
Typical defaults; always verify specific versions and flags for your environment.
| Language | Division rule for % | Remainder sign when b > 0 | Example: -13 % 5 |
|---|---|---|---|
| PHP | Truncates toward zero | Same as dividend (may be negative) | -3 |
| JavaScript | Truncates toward zero | Same as dividend (may be negative) | -3 |
| Java | Truncates toward zero | Same as dividend (may be negative) | -3 |
| C / C++ | Typically truncates toward zero | Same as dividend (may be negative) | -3 |
| Python | Floor division pair (// and % linked) | Always non‑negative if b > 0 | 2 |
Note: Many languages define a = bq + r, but choose different q (truncated vs floored). That choice determines the sign of r.
Formulas Used
- Truncated Division:
q = (int)(a / b),r = a − b × q. - Euclidean Division: find integers
qandrsuch thata = bq + rand0 ≤ r < |b|. - Relationship: The two conventions match when
a / b ≥ 0or whenr_trunc ≥ 0. - Exact Quotient:
a / b, rounded to your selected decimal places.
How to Use
- Enter Dividend (a) and Divisor (b). Avoid b = 0.
- Choose decimal places for the displayed exact quotient.
- Select the remainder convention you prefer to highlight in results.
- Tick Show steps for a concise derivation.
- Press Calculate. Export the results to CSV or PDF.
FAQs
It is the unique remainder r satisfying
a = bq + r with 0 ≤ r < |b|, regardless of signs of a or b.Truncated remainder uses
q = (int)(a/b) which truncates toward zero, so the remainder can be negative when a/b is negative.Use Euclidean for number theory or modular arithmetic. Use truncated to mirror many programming languages’
% operator behavior.Yes. The exact quotient field shows the decimal result rounded to your selected precision. Remainder logic still follows the same definitions.
You may be comparing Euclidean vs truncated conventions. Ensure you’re using the same remainder definition as your reference.