What is the modulo operation?

Modulo (mod) returns the remainder after division. a mod b = remainder when a is divided by b. Example: 17 mod 5 = 2, because 17 = 5×3 + 2. The result is always between 0 and b-1 for positive numbers.

How does modulo work with negative numbers?

For negative numbers, there are two conventions: truncated (C/JavaScript) where sign matches dividend, and floored (Python) where sign matches divisor. Example: -7 mod 3 = -1 (truncated) or 2 (floored). Our calculator shows both.

What is modular arithmetic used for?

Modular arithmetic is used in: cryptography (RSA encryption), computer science (hash functions, circular arrays), time calculations (12-hour clock), check digits (ISBN, credit cards), and number theory.

What is the difference between mod and remainder?

For positive numbers, mod and remainder are the same. For negative numbers, they may differ: remainder keeps the dividend's sign, while mathematical mod is always non-negative. -7 remainder 3 = -1, but -7 mod 3 = 2 mathematically.

What are the properties of modulo?

Key properties: (a + b) mod n = ((a mod n) + (b mod n)) mod n, (a × b) mod n = ((a mod n) × (b mod n)) mod n, a mod 1 = 0 for any integer a, a mod a = 0.

Modulo Calculator - Calculate Remainder (Mod)

Modulo Calculator

Calculate modulo operations (a mod b) with step-by-step explanations. Perfect for programmers, math students, and anyone working with remainder calculations.

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Modulo Operation Calculator

Calculate the remainder of integer division (a mod b)

Formula

a \bmod b = a - b \times \lfloor a / b \rfloor

Where a is the dividend and b is the divisor

Dividend (a) *

The number being divided (can be negative)

Divisor (b) *

The number to divide by (cannot be zero)

17 \bmod 5 = ?

Example Calculations

Common modulo operations and their results

17 mod 5

Basic positive modulo

Result: 2

13 mod 12

Clock analogy (1 o'clock)

Result: 1

25 mod 12

Multiple circles

Result: 1

42 mod 10

Programming example

Result: 2

1000 mod 7

Large number modulo

Result: 6

8 mod 3

Small divisor

Result: 2

What is Modulo Operation?

The modulo operation finds the remainder after division of one number by another. It's denoted as a mod b or a % b in programming.

Key Properties:

Result is always 0 ≤ r < |b| for positive divisor
a mod b = a - b × floor(a / b)
If a is divisible by b, then a mod b = 0
The sign of the result depends on the divisor

Applications:

Programming: Array indexing, hash functions
Mathematics: Number theory, cryptography
Real world: Clock arithmetic, calendar calculations

How to Calculate Modulo by Hand

For Positive Numbers:

Divide a by b using integer division
Find the quotient (whole number part)
Multiply quotient by divisor
Subtract from original dividend

For Negative Numbers:

Use the formula: a mod b = a - b × floor(a / b)
Calculate floor(a / b) carefully
floor(-5/2) = floor(-2.5) = -3
-5 mod 2 = -5 - 2×(-3) = -5 + 6 = 1

Remember: The mathematical definition ensures the result is always non-negative when the divisor is positive.

Modulo in Programming & Math

Programming

•Array index wrapping: i % array.length
•Even/odd check: n % 2 == 0
•Circular buffer: (index + 1) % bufferSize
•Hash table indexing: hash % tableSize

Mathematics

•Modular arithmetic: (a + b) mod m
•Cryptography: RSA encryption
•Number theory: GCD calculations
•Congruence relations: a ≡ b (mod m)

Real World

•Clock arithmetic: (hour + n) % 12
•Day of week: day % 7
•Circular patterns: step % cycle
•Remainder calculations

Modulo Strategy Guide for Math and Coding

Quick answer: modulo gives you the leftover part of division, which makes it perfect for cyclic logic. If you need wrap-around behavior (clock time, rotating UI tabs, circular buffers, week-day calculations), modulo is usually the first operator to reach for.

In modular arithmetic, numbers repeat after the modulus. For example, with modulus 12, values 0 and 12 are equivalent, as are 1 and 13. This equivalence is why modular math is foundational in scheduling, cryptography, and hashing. It compresses an unbounded integer line into a predictable cycle.

Example: suppose your app displays items in pages of 8 cards. If the global item index is 27, the in-page slot is $27\bmod 8 = 3$ . That means item 27 appears in slot 4 (zero-based index 3). Another example: if a recurring task runs every 14 days, checking $\text{daysElapsed}\bmod 14$ quickly tells you whether you're on run day (remainder 0) without date-heavy logic.

The biggest pitfall is negative numbers. Some languages return negative remainders, others return non-negative modulo values. If you're building cross-language systems, normalize with $((a\%b)+b)\%b$ when $b>0$ so results stay inside $[0,b-1]$ . This avoids off-by-one bugs in arrays, game loops, and encryption code.

For manual checks, always verify with $a=bq+r$ . If remainder $r$ falls outside the expected range, either the quotient or sign handling is wrong. Practicing this verification step makes debugging modulo-heavy algorithms much faster.

Learn More About Modular Arithmetic

Frequently Asked Questions

Modulo finds the remainder after division. a mod b = remainder when a ÷ b. Example: 17 mod 5 = 2 because 17 = 5×3 + 2. In programming, it's written as a % b.

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Key Properties:

Applications:

For Positive Numbers:

For Negative Numbers:

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