Calculate modulo (remainder) of any division with quotient and step-by-step solution.
The modulo operation (mod) returns the remainder after dividing one number by another. For example, 17 mod 5 = 2 because 17 ÷ 5 = 3 remainder 2. Written as 17 % 5 = 2 in programming. Our modulo calculator instantly computes the remainder for any two numbers, supporting both positive and negative values.
The modulo operator (%) is one of the most used operators in programming. Common uses: Even/Odd check: if (n % 2 == 0) → even. Cycling through array indices: index = count % array.length. Clock arithmetic: hour = (currentHour + hoursAdded) % 24. Generating patterns: alternating colors, striped rows in tables. Hash functions: hash % tableSize distributes data evenly.
In mathematics, modular arithmetic is a system where numbers "wrap around" after reaching a certain value (the modulus). Clock time is the classic example: 11 + 3 = 14, but 14 mod 12 = 2 (2 PM). Modular arithmetic is fundamental in number theory, cryptography, and solving problems like: "What day of the week will it be in 100 days?"
Modular arithmetic is the mathematical backbone of modern cryptography. RSA encryption uses modular exponentiation. The Diffie-Hellman key exchange is based on discrete logarithms modulo a prime. The reason these systems are secure is that while computing a^b mod n is fast, reversing the operation (finding b given a, a^b mod n, and n) is computationally infeasible for large primes.
Modulo behavior with negative numbers varies by programming language. In Python: -7 % 3 = 2 (result always non-negative, matches mathematical definition). In JavaScript/Java/C++: -7 % 3 = -1 (result has same sign as dividend). When working across languages, be explicit about expected behavior with negative inputs to avoid subtle bugs in your code.
The modulo operation returns the remainder after dividing one number by another — 17 modulo 5 equals 2, since 17 divided by 5 is 3 with a remainder of 2. While this might initially seem like a narrow mathematical curiosity, modulo is used constantly in programming and computer science for tasks like determining whether a number is even or odd (checking if a number modulo 2 equals 0 or 1), cycling through a fixed range of values (like wrapping array indices or clock hours), and implementing hash functions used throughout computer science for data organization and retrieval.
Modulo also underlies clock arithmetic in everyday life — calculating what time it will be 20 hours from 3 PM uses the same underlying logic as a modulo operation, wrapping the result back around once it exceeds the 12 or 24-hour range, which is why modulo is sometimes informally called "clock math."
Modulo behavior with negative numbers varies between programming languages and mathematical conventions in a way that surprises many people — some systems return a negative remainder for a negative dividend, while others always return a positive remainder regardless of sign, following a mathematical convention called the "Euclidean" definition of modulo. This inconsistency across systems is a genuine, well-documented source of subtle bugs in software that handles negative number modulo operations without accounting for this variation.
Modulo arithmetic quietly underlies many everyday scheduling calculations — determining which day of the week a date falls on, calculating rotating shift schedules, or figuring out which position in a repeating cycle (like a 7-day or 28-day pattern) a given day represents. Recognizing these as modulo problems, rather than solving them through manual counting, often leads to a faster and more reliable calculation.
Modulo (mod) is the remainder after dividing one number by another. For example, 17 mod 5 = 2 because 17 ÷ 5 = 3 with remainder 2.
Modulo is widely used in programming for checking even/odd numbers, cycling through arrays, clock arithmetic, and cryptography.
0 mod n = 0 for any non-zero n, because 0 divided by any number gives 0 with no remainder.
Modulo with negative numbers varies by programming language. In mathematics, the result is always non-negative. In Python, -7 mod 3 = 2, not -1.
