Modulo Calculator

The modulo operation finds the remainder when one number is divided by another. For example, 17 mod 5 = 2, because when 17 is divided by 5, the remainder is 2. The modulo operation is also known as the remainder operation and is fundamental in number theory, computer science, and cryptography.

To calculate A mod B, divide A by B and find the remainder. The formula is: A mod B = A - (B × ⌊A/B⌋), where ⌊A/B⌋ is the floor division (integer division) of A by B. For example, to calculate 23 mod 7: 23 ÷ 7 = 3.285..., floor division gives 3, so 23 mod 7 = 23 - (7 × 3) = 23 - 21 = 2.

In modulo operations, the dividend is the number being divided (A in A mod B), and the divisor is the number you're dividing by (B in A mod B). The result is the remainder after division. For example, in 17 mod 5, 17 is the dividend and 5 is the divisor, giving a remainder of 2.

This modulo calculator is perfect for programming tasks, cryptography, clock arithmetic, determining even/odd numbers, hash functions, cyclic operations, and any mathematical operations that require finding remainders. It's essential for computer science, discrete mathematics, and number theory applications.

Yes, the calculator supports negative numbers. The result follows the mathematical definition where the remainder has the same sign as the divisor (B). For example, -17 mod 5 = 3, and 17 mod -5 = -3. This is important for programming languages and mathematical consistency.

Division by zero is undefined in mathematics, so the calculator will show an error when the divisor (B) is zero. Always ensure the divisor is a non-zero number. This is a fundamental mathematical rule that applies to all division and modulo operations.

Modulo is commonly used for: determining if a number is even (n mod 2 = 0), wrapping around arrays (index mod array_length), clock arithmetic (time mod 12), hash table implementations, cyclic buffers, random number generation, and many other applications in algorithms and data structures.

Modulo operations are used in: cryptography and encryption algorithms, digital signatures, hash functions, cyclic redundancy checks (CRC), time calculations (12-hour and 24-hour clocks), calendar calculations, music theory (octave relationships), and many other practical applications in science and engineering.

In number theory, two numbers are congruent modulo n if they have the same remainder when divided by n. This is written as a ≡ b (mod n). For example, 17 ≡ 2 (mod 5) because both 17 and 2 have remainder 2 when divided by 5. This concept is fundamental in cryptography and abstract algebra.