Modulo Calculator
Calculate remainders of division operations and explore modular arithmetic. Essential for computer science, programming, cryptography, and mathematical calculations.
Calculate Modulo
Find the remainder when one number is divided by another:
a mod b = remainder of a ÷ b
a = b × quotient + remainder
0 ≤ remainder < |b|
Understanding Modular Arithmetic
Modular arithmetic is a fundamental concept in mathematics and computer science. It deals with remainders of division and has numerous applications in programming, cryptography, and digital systems.
Modulo Fundamentals
🔢 What is Modulo?
a mod b = remainder when a is divided by b
Also written as a ≡ r (mod b)
Range: 0 ≤ r < |b|
Works with negative numbers
Fundamental to computer arithmetic
🧮 Basic Operations
Addition: (a + b) mod m = (a mod m + b mod m) mod m
Multiplication: (a × b) mod m = (a mod m × b mod m) mod m
Properties depend on modulus
Used in cryptographic algorithms
🔑 Modular Inverse
Find x such that (a × x) mod m = 1
Only exists if gcd(a, m) = 1
Essential for RSA cryptography
Used in digital signatures
Computer Science Applications
💻 Programming Basics
Even/odd detection: x mod 2
Array indexing with wraparound
Circular buffer implementation
Hash table collision resolution
Random number generation
🔐 Cryptography
RSA encryption algorithm
Diffie-Hellman key exchange
Digital signature verification
Hash function implementation
Block cipher modes
🌐 Computer Systems
Memory address calculations
Cache line alignment
Thread scheduling algorithms
Network packet sequencing
File system block allocation
🎮 Game Development
Procedural content generation
Random level design
Character stat calculations
Inventory management systems
Save game checksums
Practical Examples
⏰ Time Calculations
Clock arithmetic: 15 mod 12 = 3
Day of week: date mod 7
Time zone conversions
Calendar date calculations
Alarm scheduling
📊 Data Processing
Load balancing: request mod servers
Data partitioning
Hash-based routing
Distributed computing
Cache key generation
🎨 Digital Media
Color wheel calculations
Audio sample wrapping
Video frame numbering
Animation loop timing
Pattern generation
🏦 Financial Systems
Check digit calculations
Account number validation
Interest rate computations
Loan amortization
Credit score algorithms
Common Modulo Operations
| Operation | Example | Result | Application |
|---|---|---|---|
| 17 mod 5 | 17 ÷ 5 = 3 remainder 2 | 2 | Basic division |
| -7 mod 4 | Ensure positive remainder | 1 | Negative numbers |
| 24 mod 7 | 24 ÷ 7 = 3 remainder 3 | 3 | Day of week |
| 100 mod 10 | 100 ÷ 10 = 10 remainder 0 | 0 | Multiples check |
| 15 mod 12 | 15 ÷ 12 = 1 remainder 3 | 3 | Clock arithmetic |
🔢 Modulo Tip: In programming, the modulo operator (%) gives the remainder of division. It's essential for implementing circular behavior, like wrapping around in arrays or calculating positions in cycles.
Advanced Modular Concepts
🔍 Euclidean Algorithm
Greatest common divisor calculation
Extended Euclidean algorithm
Used for modular inverse
Efficient GCD computation
Basis for cryptography
🎲 Pseudorandom Numbers
Linear congruential generators
Mersenne Twister algorithm
Modulo-based randomization
Game random number generation
Simulation and modeling
🔐 Number Theory
Fermat's Little Theorem
Euler's Totient function
Chinese Remainder Theorem
Primality testing algorithms
Factorization methods
💽 Error Detection
Checksum calculations
Cyclic redundancy checks
ISBN validation
Credit card number verification
Data integrity algorithms
Programming Language Examples
🐍 Python
result = a % b
Built-in modulo operator
Works with negative numbers
Large integer support
Consistent behavior
☕ JavaScript
result = a % b
Same as Python behavior
Follows IEEE 754 standard
Works with floating point
Type coercion handling
⚙️ C/C++
result = a % b
Sign follows dividend
Integer division behavior
Undefined for negative divisors
Processor-specific implementation
🦀 Rust
result = a % b
Sign follows dividend
Checked arithmetic available
Consistent with C/C++
Memory safety guarantees