Analyze modular results, quotients, residues, and patterns. Test operations, compare outputs, and review worked examples. Understand modular behavior using exports, graphs, and simple steps.
Field guide:
The graph shows the Euclidean remainder pattern for x mod n across the selected integer interval.
| Case | Expression | Result | Notes |
|---|---|---|---|
| Basic remainder | 29 mod 6 | 5 | 29 = 6 × 4 + 5 |
| Euclidean mod | -29 mod 6 | 1 | Euclidean remainder stays non negative. |
| Modular addition | (17 + 25) mod 9 | 6 | 42 mod 9 = 6 |
| Modular multiplication | (11 × 13) mod 7 | 3 | 143 mod 7 = 3 |
| Modular power | 5^4 mod 13 | 1 | 625 mod 13 = 1 |
| Modular inverse | 3^-1 mod 11 | 4 | 3 × 4 ≡ 1 (mod 11) |
Basic remainder: a = n × q + r, where the remainder r comes from division by n.
Euclidean mod: mod(a, n) = ((a % n) + n) % n. This keeps the result between 0 and n − 1 for positive n.
Addition: (a + b) mod n = ((a mod n) + (b mod n)) mod n.
Subtraction: (a − b) mod n = ((a mod n) − (b mod n)) mod n.
Multiplication: (a × b) mod n = ((a mod n) × (b mod n)) mod n.
Power: a^k mod n is computed efficiently with repeated squaring.
Inverse: a has an inverse modulo n only when gcd(a, n) = 1.
Division: a / b mod n = a × b^-1 mod n, if b^-1 exists modulo n.
The mod function measures what remains after division. It is one of the most useful tools in number theory, discrete mathematics, cryptography, hashing, scheduling, coding theory, and computer science. When you reduce values modulo n, large integers collapse into a fixed residue system. That makes patterns easier to study and calculations easier to manage.
This page does more than a simple remainder check. It handles standard remainder output, Euclidean mod, modular addition, subtraction, multiplication, exponentiation, inverse testing, and modular division when an inverse exists. The Euclidean option is especially helpful because it returns a non negative remainder, which is the preferred form in many mathematics courses and algorithms.
The graph displays x mod n across an interval you choose. This visual makes periodic structure obvious. You can see the repeating staircase pattern, detect residue cycles quickly, and compare how the modulus changes the sequence. It is a useful teaching aid for classroom demonstrations, homework checks, and pattern recognition.
Use the basic remainder mode when you want the direct output from division. Use Euclidean mod when you need residues in the range from 0 to n − 1. Use power mode for congruence problems and fast exponentiation practice. Use inverse and division when solving modular equations, simplifying congruences, or checking whether a value is invertible under a chosen modulus.
This calculator also keeps a session history, provides a sample data table, and supports export tools for quick reporting. That makes it useful for students, teachers, tutors, and developers who need both accurate outputs and an organized workflow while studying modular relationships.
Mod gives the remainder after division. In modular arithmetic, it also represents an equivalence class, where numbers with the same remainder are treated as congruent under the same modulus.
Euclidean mod adjusts the raw remainder into the standard non negative range. That is useful because many proofs, algorithms, and classroom definitions prefer residues from 0 up to n − 1.
A modular inverse exists only when the number and the modulus are coprime. In other words, gcd(a, n) must equal 1. Otherwise no multiplicative inverse exists modulo n.
Modular powers appear in cryptography, congruence problems, primality work, and algorithm design. Repeated squaring computes large powers efficiently without expanding huge intermediate numbers.
Yes, but only indirectly. You multiply by the modular inverse of the divisor. If that inverse does not exist under the chosen modulus, modular division is not defined.
The values repeat because residues cycle every n steps. Once x increases by the modulus, the remainder pattern restarts, producing the familiar periodic modular structure.
Use the raw remainder when you want the direct division output. Use Euclidean mod when you need standard residues for proofs, programming tasks, or modular algebra practice.
Yes. It is useful for learning residues, inverses, exponentiation, and congruences. Those ideas are foundational in many introductory cryptography topics and modular number theory exercises.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.