Modulo Calculator
Compute the modulo operation (remainder of an integer division)
Compute modulo
What is the modulo operation?
The modulo operation computes the remainder of a division. For a mod b, a is divided by b and the remaining remainder is returned.
Modulo Info
Properties
Modulo: Remainder of an integer division
Range: 0 ≤ (a mod b) < |b| for b ≠ 0
Notation: a mod b or a % b
Quick examples
Formulas and definitions of the modulo operation
Definition
r is the unique remainder of dividing a by b
Division algorithm
Euclidean division algorithm with quotient q and remainder r
Properties
Distributivity of the modulo operation
Negative numbers
Handling of negative dividends
Congruence
Equivalence of modulo and congruence
Special cases
Important special cases of the modulo operation
Example calculations with modulo
Example 1: 11 mod 4
Example 2: 17 mod 5
Example 3: Negative number (-7 mod 3)
Example 4: Perfect division (10 mod 2)
Modulo properties demonstrated
The modulo operation follows mathematical laws
Applications of the modulo operation
The modulo operation has many practical and theoretical applications:
Programming & computer science
- Hash functions and hash tables
- Pseudo-random number generators
- Circular buffers and ring buffers
- Array indexing (wrap-around)
Cryptography & security
- RSA encryption
- Modular arithmetic
- Diffie-Hellman key exchange
- Digital signatures
Timekeeping & cycles
- Compute weekdays (mod 7)
- 12-hour vs 24-hour format
- Calendar calculations
- Periodic events
Mathematics & number theory
- Congruence arithmetic
- Divisibility tests
- Chinese remainder theorem
- Discrete mathematics
The modulo operation: Foundation of number theory
The modulo operation is far more than just a "remainder of a division." It forms the foundation of modern number theory and enables elegant solutions to complex problems. From simple checks for even/odd numbers to highly secure encryption schemes, the modulo operation shows how basic mathematical concepts find practical applications in the digital world. It defines equivalence classes and creates algebraic structures that are indispensable in cryptography, computer science and discrete mathematics.
Properties
- 0 ≤ (a mod n) < n for n > 0
- Distributivity over addition/multiplication
- Defines equivalence relations
- Periodic/cyclic behavior
Significance
- Basis of congruence arithmetic
- Core of cryptographic schemes
- Enables efficient algorithms
- Structures discrete mathematics
Applications
- Cryptography and IT security
- Hash functions and data structures
- Time and calendar calculations
- Random numbers and simulations
Summary
The modulo operation links elementary arithmetic with cutting-edge applications. What begins as a simple "remainder of a division" becomes a powerful tool of discrete mathematics. From detecting even numbers (n mod 2 = 0) to RSA encryption, the modulo operation demonstrates how mathematical elegance and practical utility harmonize. It exemplifies how simple concepts lead to complex and secure systems.