Enter Values
Example Data (click to fill)
| Example | Mode | Use |
|---|---|---|
| 98765 ÷ 123 | euclidean | |
| 255 ÷ 16 | euclidean | |
| -137 ÷ 12 | euclidean | |
| 37 ÷ -5 | euclidean | |
| 1000 ÷ 7 (truncated) | truncated | |
| 0 ÷ 9 | euclidean |
Result
Enter values and press Calculate to see results.
Formula Used
For integers a (dividend) and b (divisor ≠ 0), we find integers q and r such that a = b·q + r.
- Euclidean convention: remainder r is unique with 0 ≤ r < |b|. Quotient is q = (a - r)/b.
- Truncated convention: q = trunc(a/b) and r = a - b·q; here r may be negative.
How to Use This Calculator
- Enter integer values for dividend and divisor (divisor ≠ 0).
- Select a remainder convention: Euclidean (non‑negative remainder) or Truncated.
- Optionally enable Show long division steps for digit‑by‑digit working.
- Press Calculate to display quotient, remainder, and identity check.
- Use Download CSV to export results and detailed steps; use Save as PDF to print or save.
FAQs
Euclidean guarantees a non‑negative remainder r with 0 ≤ r < |b|. Truncated uses a quotient truncated toward zero; the remainder can be negative.
Yes. The algorithm handles all integer signs. For steps, we divide absolute values and apply the sign to the final quotient as needed.
Because we compute q and r from exact integer arithmetic, then verify by reconstruction: a - (b·q + r) = 0 within integer arithmetic.
Typical 64‑bit integer ranges work well. The step simulation avoids overflow by using per‑digit operations on strings for the dividend.
Yes. Use Download CSV for spreadsheets. For documents, use Save as PDF to print the result view.
Example Data Table
| Dividend | Divisor | Mode | Quotient | Remainder |
|---|---|---|---|---|
| 98765 | 123 | euclidean | 802 | 119 |
| 255 | 16 | euclidean | 15 | 15 |
| -137 | 12 | euclidean | -12 | 7 |
| 37 | -5 | euclidean | -7 | 2 |
| 1000 | 7 | truncated | 142 | 6 |
| 0 | 9 | euclidean | 0 | 0 |
Remainder Conventions Compared (Worked Examples)
See how truncated and Euclidean conventions differ for the same inputs. This clarifies sign behavior and remainder ranges.
| Dividend (a) | Divisor (b) | Truncated q | Truncated r | Euclidean q | Euclidean r |
|---|---|---|---|---|---|
| -37 | 5 | -7 | -2 | -8 | 3 |
| 37 | -5 | -7 | 2 | -7 | 2 |
| 255 | 16 | 15 | 15 | 15 | 15 |
| 1000 | 7 | 142 | 6 | 142 | 6 |
| -137 | 12 | -11 | -5 | -12 | 7 |
| 123456 | 89 | 1387 | 13 | 1387 | 13 |
Reminder: Euclidean remainder satisfies 0 ≤ r < |b|, while truncated can yield negative remainders.
Divisibility Quick Reference (Remainder Rules)
Handy tests and remainder ranges used when planning or checking long division operations.
| Divisor | Quick Test | Possible Remainders (Euclidean) |
|---|---|---|
| 2 | Last digit even | 0 or 1 |
| 3 | Sum of digits divisible by 3 | 0,1,2 |
| 4 | Last two digits divisible by 4 | 0–3 |
| 5 | Ends with 0 or 5 | 0–4 |
| 6 | Divisible by 2 and 3 | 0–5 |
| 8 | Last three digits divisible by 8 | 0–7 |
| 9 | Sum of digits divisible by 9 | 0–8 |
| 10 | Ends with 0 | 0–9 |
| 11 | Alternating digit sum rule | 0–10 |
| 12 | Divisible by 3 and 4 | 0–11 |
Euclidean remainders always lie in [0, |b|). Truncated remainders may be negative.