Modulo Calculator

Calculate remainders with step-by-step solutions, modular arithmetic operations, negative number handling, and real-world applications.

Formula:a mod b = remainder

Result

Remainder (mod)

2

Dividend17
Divisor5
Quotient3

Calculate Modulo

The number to be divided

The number to divide by (non-zero)

17 mod 5 = 2

17 = 3 x 5 + 2

Step-by-Step Solution

1

Given: 17 mod 5

2

Formula: a = q x b + r, where r is the remainder

4

Step 1: Divide 17 by 5

5

17 / 5 = 3.400000

7

Step 2: Take the integer part (quotient)

8

Quotient (q) = 3

10

Step 3: Calculate remainder

11

Remainder (r) = 17 - (3 x 5)

12

Remainder (r) = 17 - 15

13

Remainder (r) = 2

15

Result: 17 mod 5 = 2

Verification: 3 x 5 + 2 = 17 = 17

Result

Remainder (mod)

2

Dividend17
Divisor5
Quotient3

?What is Modulo?

The modulo operation (a mod b) finds the remainder when dividing a by b. For example, 17 mod 5 = 2 because 17 = 3 x 5 + 2. The formula is: a = q x b + r, where q is the quotient and r is the remainder. Modulo is used in clock arithmetic (13:00 mod 12 = 1), day calculations, cryptography, hash functions, and checking divisibility.

What is the Modulo Operation?

The modulo operation finds the remainder after division of one number by another. Written as a mod b or a % b, it returns r such that a = q x b + r where 0 <= r < |b|. Modular arithmetic is fundamental in computer science (array indexing, hash functions), cryptography (RSA encryption), and everyday applications like clock time and calendar calculations.

Key Facts About Modulo

  • a mod b = remainder when a is divided by b
  • 17 mod 5 = 2 because 17 = 3 x 5 + 2
  • Clock arithmetic: 15 mod 12 = 3 (3 o'clock)
  • Day of week: (current + days) mod 7
  • If a mod b = 0, then b divides a evenly
  • Different languages handle negatives differently
  • Modular addition: (a + b) mod n
  • Used in cryptography (RSA) and checksums

Frequently Asked Questions

The modulo operation (mod) finds the remainder when dividing one integer by another. For example, 17 mod 5 = 2 because 17 = 3 x 5 + 2, so the remainder is 2.
Different conventions exist. In JavaScript/C/Java (truncated division), -7 mod 3 = -1. In Python (floored division), -7 mod 3 = 2. The Euclidean definition always returns a non-negative remainder.
Clock arithmetic is modulo 12 (or 24). If it's 10 o'clock and you add 5 hours, you get (10 + 5) mod 12 = 3 o'clock. This is why 13:00 = 1:00 PM.
Modular arithmetic is fundamental to RSA and other encryption algorithms. It provides one-way functions that are easy to compute forward but hard to reverse, making encryption secure.
If a mod b = 0, then a is evenly divisible by b. For example, 15 mod 5 = 0 means 15 is divisible by 5. This is useful for checking even/odd (n mod 2), divisibility rules, and more.

Last updated: 2025-01-15