MathIsimple
Modulo Calculator

Modulo Calculator

Calculate modulo operations (a mod b) with step-by-step explanations. Perfect for programmers, math students, and anyone working with remainder calculations.

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Modulo Operation Calculator
Calculate the remainder of integer division (a mod b)
Formula
a mod b=a−b×⌊a/b⌋a \bmod b = a - b \times \lfloor a / b \rfloor

Where a is the dividend and b is the divisor

The number being divided (can be negative)

The number to divide by (cannot be zero)

Example Calculations
Common modulo operations and their results
17 mod 5 = 2
Basic positive modulo
-17 mod 5 = 3
Negative dividend
17 mod -5 = -3
Negative divisor
-17 mod -5 = -2
Both negative
1000 mod 7 = 6
Large number modulo
42 mod 10 = 2
Programming example
What is Modulo Operation?

The modulo operation finds the remainder after division of one number by another. It's denoted as a mod b or a % b in programming.

Key Properties:

  • Result is always 0 ≤ r < |b| for positive divisor
  • a mod b = a - b × floor(a / b)
  • If a is divisible by b, then a mod b = 0
  • The sign of the result depends on the divisor

Applications:

  • Programming: Array indexing, hash functions
  • Mathematics: Number theory, cryptography
  • Real world: Clock arithmetic, calendar calculations
How to Calculate Modulo by Hand

For Positive Numbers:

  1. Divide a by b using integer division
  2. Find the quotient (whole number part)
  3. Multiply quotient by divisor
  4. Subtract from original dividend

For Negative Numbers:

  1. Use the formula: a mod b = a - b × floor(a / b)
  2. Calculate floor(a / b) carefully
  3. floor(-5/2) = floor(-2.5) = -3
  4. -5 mod 2 = -5 - 2×(-3) = -5 + 6 = 1

Remember: The mathematical definition ensures the result is always non-negative when the divisor is positive.

Modulo in Programming & Math

Programming

  • •Array index wrapping: i % array.length
  • •Even/odd check: n % 2 == 0
  • •Circular buffer: (index + 1) % bufferSize
  • •Hash table indexing: hash % tableSize

Mathematics

  • •Modular arithmetic: (a + b) mod m
  • •Cryptography: RSA encryption
  • •Number theory: GCD calculations
  • •Congruence relations: a ≡ b (mod m)

Real World

  • •Clock arithmetic: (hour + n) % 12
  • •Day of week: day % 7
  • •Circular patterns: step % cycle
  • •Remainder calculations

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