Modulo Calculator (Reszta z dzielenia)

Compute modulo (remainder / reszta z dzielenia) for integer division with normalized results.

Modulo Calculator (Reszta z dzielenia)

Compute modulo (remainder / reszta z dzielenia) with normalized results and optional steps.

Inputs Integers only
The number being divided. You can use negatives (e.g., -29).
Must be non-zero.
Use normalized modulo for wrap-around logic; use remainder to match many language operators.
Result Ready
No output yet
Enter your numbers and click Calculate.
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About Modulo Calculator (Reszta z dzielenia)

Modulo Calculator (Reszta z dzielenia) – Remainder and Modulo Tool

Use this Modulo Calculator to compute the remainder (in Polish: reszta z dzielenia) from integer division in seconds. It’s built for everyday math, programming, and debugging—showing the answer in a clean format and, when you want it, the reasoning behind it.

Modulo is one of those small ideas that quietly powers big systems: repeating schedules, circular user interfaces, cryptographic building blocks, hash table bucketing, and game logic that “wraps” positions around a map. If you’ve ever asked “what’s left after dividing?” or “how do I keep a value inside a range?”, you’ve used modulo—even if you didn’t call it that.

This tool focuses on integer modulo. That choice is intentional: quotient-and-remainder arithmetic is the clearest and most widely taught definition, and it maps directly to how most programming languages implement the % operator. When decimals are involved, results depend heavily on rounding and floating-point representation, so the “simple” answer can become misleading. For stable, predictable outcomes, integer modulo is the gold standard.

How It Works

Enter two integers: the dividend (a, the number being divided) and the divisor (b, the number you divide by). The calculator computes the quotient and the remainder using standard integer arithmetic, then optionally produces a normalized modulo result that is always non‑negative.

Core definition

For integers a and non‑zero b, there exist integers q (quotient) and r (remainder) such that:

  • Identity: a = b × q + r
  • Remainder bounds (normalized): 0 ≤ r < |b|

Some programming environments use truncating division for q (rounding toward zero). In those cases, the remainder can be negative when a is negative. That’s not “wrong”—it’s simply a different convention. To avoid confusion, this tool displays both the language-style remainder and a normalized modulo result that stays within 0..|b|-1.

What you’ll see in the results panel

  • Quotient (integer division): how many whole times the divisor fits into the dividend.
  • Remainder (language-style): the direct remainder produced by the common % operator behavior.
  • Normalized modulo: a non‑negative result in the range 0..|b|-1, ideal for wrap-around logic.
  • Equation check: the identity a = b × q + r so you can verify the calculation at a glance.
  • Steps (optional): intermediate values and formulas used so you can learn or audit the result.

Quick examples

  • Positive numbers: 29 ÷ 5 gives q = 5 and r = 4 because 29 = 5×5 + 4.
  • Negative dividend: -29 ÷ 5 may produce a remainder of -4 in truncating systems, but the normalized modulo is 1 because -29 ≡ 1 (mod 5).
  • Wrap-around: If today is day i and you move back 2 days, use normalized modulo to stay in range: ((i-2) mod 7).

Key Features

Instant remainder (reszta) calculation

Type your dividend and divisor and get the remainder immediately, with no extra steps. The layout keeps inputs and output visible at the same time so you can tweak values quickly—perfect for homework checking, quick estimations, or debugging a loop condition.

Normalized modulo for stable ranges

Many real tasks expect results inside a fixed range. For example, you might need a number from 0 to 23 for hours, from 0 to 59 for minutes, or from 0 to n-1 for an array index. The normalized modulo result from this tool stays non‑negative and fits neatly into those ranges.

Clear handling of negative numbers

Negative inputs are where modulo becomes tricky. Different programming languages and libraries disagree on whether the sign of the result should follow the dividend, the divisor, or always be non‑negative. This calculator makes the ambiguity explicit by showing both the direct remainder and the normalized modulo, helping you match your target environment and avoid off-by-one errors.

Step-by-step mode for learning and auditing

Turn on steps to see the arithmetic that leads to the final remainder. This is helpful for students learning division with remainders, for teachers demonstrating why a result is correct, and for developers verifying that a “mod fix” behaves correctly for negative values.

Copy, download, and share-friendly output

The result panel is designed to be portable. Copy a compact summary to your clipboard to paste into code comments, tickets, or documentation. Download the output as a text file when you want to keep a small library of test cases for later regression testing.

Safe validation and division-by-zero protection

Modulo by zero is undefined and can crash programs. This tool validates that the divisor is not zero before computing anything, ensuring the output is always meaningful and preventing confusing “infinite” or “NaN” style states.

Use Cases

  • Array and string indexing: Keep indexes in bounds for circular access—carousels, ring buffers, and repeating animations often rely on modulo.
  • Calendars and time math: Convert large offsets into cycles like hours (mod 24), minutes (mod 60), weekdays (mod 7), or repeating shift rotations.
  • Even/odd checks and divisibility: Test parity with n mod 2, or confirm divisibility by checking whether the remainder is 0.
  • Hash buckets and partitioning: Place items into a fixed number of buckets with hash(x) mod k to distribute load.
  • Game development: Wrap positions around a grid, keep angles within 0..359, or cycle through player turns with round-robin logic.
  • Cryptography and number theory study: Learn modular arithmetic fundamentals used in RSA-style constructions, congruences, and modular inverses (without needing a full algebra system).
  • Porting code between languages: Compare the “remainder” behavior of one language to the “modulo” behavior expected in another, especially for negative inputs.

Modulo is also great for designing user-friendly limits. For example, when a user scrolls past the end of a list, you can wrap their position back to the beginning. When a counter exceeds a maximum, you can fold it back into range without writing branching logic for every edge case. When you understand modulo, cycles become easier to reason about—and bugs become easier to spot.

If you’re learning, try a few patterns: pick a divisor like 7, then test dividends 0..20 and observe how the remainders repeat. Then try negatives like -1, -2, -8 and compare the direct remainder with the normalized modulo. That simple experiment explains a lot of real-world “why is my index negative?” problems.

Optimization Tips

Choose the convention that matches your goal

If your goal is the leftover after truncating division (common in many languages), the direct remainder is appropriate. If your goal is wrap-around behavior or a value in a fixed range, the normalized modulo is usually the safer choice. When writing software, state your intent in comments or function names—“remainder” and “modulo” are not always interchangeable.

Normalize for negative offsets

Negative offsets are extremely common: moving left in a carousel, subtracting time, or applying user input that can go below zero. A normalized modulo like ((a % m) + m) % m with m = |b| ensures the result remains stable and non‑negative. That stability reduces branching logic and prevents accidental out-of-range indexing.

Validate the divisor early

Always check that the divisor is non‑zero before performing division or modulo. In robust systems, it’s better to fail early with a clear message than to allow a runtime error. If your divisor comes from user input, validation should be part of the form layer. If it comes from configuration, consider assertions or defensive checks.

Keep test cases for boundary values

Modulo bugs often appear at boundaries: when values are exactly divisible, when values are just below zero, or when the divisor changes sign. Save a small set of “known good” cases (including negative dividends) and re-check them whenever you refactor. This tool’s copy/download output makes it easy to build such a test list.

FAQ

It is the Polish phrase for the remainder after division. For example, 29 ÷ 5 leaves a remainder of 4 because 29 = 5×5 + 4.

Many environments implement % as a remainder based on truncating division, so the sign can follow the dividend. That behavior is consistent with the identity a = b×q + r when q is truncated toward zero. If you need a non‑negative result for wrap-around logic, use normalized modulo such as ((a % m) + m) % m with m = |divisor|.

Modulo by zero is undefined. This tool blocks the operation by validating that the divisor is not zero. In your own code, you should also check for 0 before using division or modulo, and decide how your application should report that error.

This calculator focuses on integers, because remainder and modulo are most consistently defined for integer division. For floating‑point mod behavior you can use language-specific functions, but be mindful that floating-point arithmetic can introduce rounding artifacts.

If you have a list length n and an index i, use normalized modulo to keep the index in range: ((i % n) + n) % n. This guarantees a result between 0 and n-1 even when i is negative.

Why Choose This Tool

Modulo is deceptively simple: one operator, multiple conventions. When you’re learning, teaching, porting code, or validating edge cases, clarity matters more than speed. This calculator shows both a language-style remainder and a normalized modulo, so you can match the result to your exact situation without guesswork.

It’s also designed for practical workflows. You can keep a library of test cases, quickly copy the output into a bug report, or download results when you’re building unit tests. With step mode, you can verify the math behind every answer and build intuition around cycles—one of the most useful patterns in both mathematics and software.