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Modulo Calculator

Calculate the modulo (remainder) of a division operation.

Enter Values

mod

Expression

17 mod 5 = 2

Remainder (Modulo)

2

17 mod 5 = 2

Division Details

Quotient (integer)3
Floor Quotient3
Exact Division3.400000
JavaScript Modulo (%)2
Mathematical Modulo2

Verification

dividend = quotient * divisor + remainder

17 = 3 * 5 + 2

17 = 17 (correct)

Modulo Operation

Definition

a mod b = a - b * floor(a / b)

The modulo operation finds the remainder after division.

Properties

0 ≤ a mod b < |b| (for mathematical modulo)

(a + b) mod n = ((a mod n) + (b mod n)) mod n

(a * b) mod n = ((a mod n) * (b mod n)) mod n

What is the Modulo Operation?

The modulo operation (also called modulus or mod) finds the remainder after dividing one number by another. It's a fundamental operation in mathematics and computer science, used in everything from cryptography to everyday calculations like determining what day of the week it will be.

Definition:

  • a mod n = the remainder when a is divided by n
  • Notation: a mod n, a % n (programming), or a (mod n)
  • Result is always between 0 and n-1 (for positive n)

Simple Examples:

Operation Division Result Explanation
17 mod 5 17 = 5×3 + 2 2 Remainder after dividing 17 by 5
24 mod 6 24 = 6×4 + 0 0 24 is divisible by 6
7 mod 10 7 = 10×0 + 7 7 7 < 10, so remainder is 7
100 mod 7 100 = 7×14 + 2 2 14 sevens fit, 2 left over

Modulo Formula and Definition

The modulo operation can be defined mathematically:

Modulo Definition

a mod n = a - n × ⌊a/n⌋ Where: • ⌊x⌋ is the floor function (round down) • a is the dividend • n is the divisor (modulus) Alternatively: a = n × q + r where q = quotient, r = remainder (a mod n) and 0 ≤ r < n Example: 23 mod 7 23 = 7 × ⌊23/7⌋ + r 23 = 7 × 3 + 2 So 23 mod 7 = 2

Where:

  • a= Dividend (number being divided)
  • n= Divisor (modulus)
  • ⌊x⌋= Floor function - greatest integer ≤ x
  • q= Quotient (integer result of division)
  • r= Remainder (mod result)

Properties of Modular Arithmetic

Modular arithmetic follows specific rules that make calculations easier:

Modular Arithmetic Properties

Addition: (a + b) mod n = ((a mod n) + (b mod n)) mod n Subtraction: (a - b) mod n = ((a mod n) - (b mod n) + n) mod n Multiplication: (a × b) mod n = ((a mod n) × (b mod n)) mod n Power: a^k mod n = (a mod n)^k mod n Examples: (17 + 8) mod 5 = (2 + 3) mod 5 = 5 mod 5 = 0 (17 × 8) mod 5 = (2 × 3) mod 5 = 6 mod 5 = 1

Where:

  • a, b= Any integers
  • n= The modulus
  • mod= Modulo operation

Negative Numbers and Modulo

Handling negative numbers in modulo varies by definition:

Definition -7 mod 3 Used In
Truncated Division -1 C, C++, Java, JavaScript
Floored Division 2 Python, Ruby, Mathematics
Euclidean 2 Mathematical definition

To always get a positive result:

((a mod n) + n) mod n works in any language

Example: -7 mod 3

  • Truncated: -7 = 3 × (-2) + (-1), so result is -1
  • Floored: -7 = 3 × (-3) + 2, so result is 2

How to Use This Calculator

Our modulo calculator handles various operations:

  1. Enter Dividend: The number being divided (a)
  2. Enter Divisor: The modulus (n)
  3. Click Calculate: Get the remainder
  4. View Results:
    • Remainder (mod result)
    • Quotient (integer division)
    • Full equation: a = n × q + r
    • Both truncated and floored results

Additional Features:

  • Modular arithmetic calculator
  • Modular exponentiation (a^b mod n)
  • Modular inverse finder
  • Congruence checker

Input Types:

  • Positive integers: 17 mod 5
  • Negative integers: -7 mod 3
  • Large numbers: 123456789 mod 97

Applications of Modulo

Modulo is essential in many practical applications:

Time and Calendars:

  • Clock arithmetic: (hour + x) mod 12 or mod 24
  • Day of week: (today + days) mod 7
  • Leap years: year mod 4 = 0 (simplified)

Computer Science:

  • Hash functions: index = hash mod table_size
  • Circular buffers: pos = (pos + 1) mod size
  • Random numbers: Many generators use modulo
  • Array indexing: Wrap-around behavior

Cryptography:

  • RSA encryption: c = m^e mod n
  • Diffie-Hellman: Key exchange
  • Hash functions: MD5, SHA use modular arithmetic
  • Checksums: ISBN, credit card validation

Everyday Uses:

  • Even/odd check: n mod 2 (0 = even, 1 = odd)
  • Digit extraction: n mod 10 gets last digit
  • Divisibility: a mod b = 0 means b divides a
  • Cycling through options: next = (current + 1) mod total

Modular Multiplicative Inverse

The modular inverse of a (mod n) is a number b such that (a × b) mod n = 1:

Modular Inverse

a × a⁻¹ ≡ 1 (mod n) Exists if and only if GCD(a, n) = 1 Finding Methods: 1. Extended Euclidean Algorithm 2. Fermat's Little Theorem (when n is prime): a⁻¹ ≡ a^(n-2) (mod n) Example: 3⁻¹ mod 7 3 × ? ≡ 1 (mod 7) 3 × 5 = 15 = 7×2 + 1 ≡ 1 (mod 7) So 3⁻¹ ≡ 5 (mod 7)

Where:

  • a⁻¹= Modular inverse of a
  • ≡= Congruent (equal under mod)
  • GCD= Greatest Common Divisor

Worked Examples

Basic Modulo Calculation

Problem:

Calculate 47 mod 8

Solution Steps:

  1. 1Divide 47 by 8: 47 ÷ 8 = 5.875
  2. 2Integer part (quotient) = 5
  3. 3Quotient × divisor = 5 × 8 = 40
  4. 4Remainder = 47 - 40 = 7
  5. 5Verify: 8 × 5 + 7 = 40 + 7 = 47 ✓

Result:

47 mod 8 = 7

Clock Arithmetic

Problem:

It's 10 AM. What time will it be in 27 hours?

Solution Steps:

  1. 1Current hour: 10
  2. 2Add hours: 10 + 27 = 37
  3. 3Apply mod 24: 37 mod 24
  4. 437 = 24 × 1 + 13
  5. 5Result: 13 (1 PM on a 24-hour clock)
  6. 6Or: (37 mod 12) = 1 (1 o'clock on 12-hour)

Result:

1 PM (13:00)

Day of Week Calculation

Problem:

Today is Tuesday (day 2). What day will it be in 100 days?

Solution Steps:

  1. 1Number the days: Sun=0, Mon=1, Tue=2, ..., Sat=6
  2. 2Today = 2 (Tuesday)
  3. 3Add days: 2 + 100 = 102
  4. 4Apply mod 7: 102 mod 7
  5. 5102 = 7 × 14 + 4
  6. 6Result: 4 = Thursday

Result:

Thursday

Tips & Best Practices

  • ✓a mod n is always between 0 and n-1 (for positive n in most definitions)
  • ✓If a < n, then a mod n = a (nothing to divide)
  • ✓a mod n = 0 means n divides a evenly (a is divisible by n)
  • ✓n mod 2 is 0 for even numbers, 1 for odd numbers
  • ✓n mod 10 gives the last digit of n
  • ✓For negative numbers: ((a mod n) + n) mod n always gives positive result
  • ✓Modular arithmetic preserves operations: (a + b) mod n = ((a mod n) + (b mod n)) mod n

Related Calculators

Exponent Calculator

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Prime Factorization Calculator

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Scientific Calculator

Advanced calculations

Binary Calculator

Base conversion

Frequently Asked Questions

Different programming languages use different definitions for the division operation that defines modulo. Languages like C, Java, and JavaScript use 'truncated division' (round toward zero), giving -7 % 3 = -1. Python uses 'floored division' (round down), giving -7 % 3 = 2. Both are mathematically valid, just different conventions. The floored version always gives positive results for positive divisors.
For positive numbers, they're identical. For negative numbers, 'mod' (mathematical modulo) typically gives a result with the same sign as the divisor, while 'rem' (remainder) gives a result with the same sign as the dividend. In math: -7 mod 3 = 2. In programming (rem): -7 rem 3 = -1. Python's % is true mod; C's % is rem.
Modular arithmetic is 'one-way' - easy to compute but hard to reverse. If you know a, b, and n, computing a^b mod n is fast. But given only the result and n, finding a and b is extremely hard (the discrete logarithm problem). This asymmetry is the foundation of RSA, Diffie-Hellman, and many other cryptographic systems.
Use modulo: a is divisible by b if a mod b = 0. For example: 24 mod 6 = 0, so 24 is divisible by 6. 25 mod 6 = 1, so 25 is not divisible by 6. This is useful for checking even/odd (n mod 2), divisibility by 3, 5, 9, etc.
Modular exponentiation calculates a^b mod n efficiently. Instead of computing a^b (which could be astronomically large) then taking mod, we use repeated squaring with mod at each step. This keeps numbers manageable. For example, 7^100 mod 13 can be computed quickly without ever calculating 7^100 directly. This is essential for RSA encryption.
Mathematically, yes, but it's rarely used and behavior varies by language. Most practical applications use positive moduli. If needed, |n| is typically used. The important thing is that the result's range depends on the definition: truncated gives results between -(|n|-1) and |n|-1, floored keeps the sign of n.

Sources & References

Last updated: 2026-01-22

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Editorial Note

MyCalcBuddy Editorial Team

This page is maintained as an educational calculator reference.

Source

Formula Source: Handbook of Mathematical Functions

by Abramowitz & Stegun

UpdatedLast reviewed: May 2026
CheckedFormula checks are based on standard references and internal QA review.

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