Modulus Calculator
Modulus - Solve mathematical problems with step-by-step solutions.
Modulus Calculator
Find the Remainder of a Division
Modulus Operation
The modulus operation (often abbreviated as 'mod') finds the remainder after the division of one number by another. For example, 10 mod 3 is 1, because 10 divided by 3 is 3 with a remainder of 1.
How the Modulus Calculator Works
The modulus (mod) operation finds the remainder after division. For example, 17 mod 5 = 2 because 17 ÷ 5 = 3 with a remainder of 2. It's written as "a mod n" or "a % n" in programming languages.
Understanding the Result
The modulus operation always returns a value between 0 and n-1 (where n is the divisor). This makes it useful for creating repeating cycles, checking even/odd numbers, and many programming applications.
How Modulus Works
Step-by-Step Process
- Divide a by n (ignore decimals, just the whole number quotient)
- Multiply the quotient by n
- Subtract that result from a to get the remainder
Examples
17 mod 5:
17 ÷ 5 = 3 (quotient)
3 × 5 = 15
17 - 15 = 2 (remainder)
25 mod 7:
25 ÷ 7 = 3 (quotient)
3 × 7 = 21
25 - 21 = 4 (remainder)
10 mod 2:
10 ÷ 2 = 5 (quotient)
5 × 2 = 10
10 - 10 = 0 (remainder - evenly divisible)
Common Patterns
Even or Odd Test
n mod 2 = 0: n is even
n mod 2 = 1: n is odd
This is the most common use of modulus in everyday programming.
Cycling Through Values
Modulus creates repeating cycles. For example, with mod 7 (days of the week):
0 mod 7 = 0, 1 mod 7 = 1, ..., 6 mod 7 = 6
7 mod 7 = 0, 8 mod 7 = 1, ..., 13 mod 7 = 6
The pattern repeats every 7 numbers, perfect for calendar calculations.
Wraparound Behavior
When counting with limits, modulus handles wraparound automatically. Clock arithmetic uses mod 12 (or mod 24) - for example, 15:00 + 10 hours = 1:00, which is (15 + 10) mod 24 = 1.
Real-World Applications
Programming
Array indexing, hash tables, circular buffers, and game development (rotating through states). Modulus is one of the most frequently used operations in computer science.
Cryptography
Essential for encryption algorithms like RSA, Diffie-Hellman, and hash functions. Modular arithmetic forms the mathematical foundation of modern cryptography.
Calendar Calculations
Determine day of week, handle time zones, and calculate recurring events. Any time you need to wrap around a fixed cycle, modulus is the answer.
Check Digits
Validate credit card numbers, ISBNs, and UPC codes using modulus-based algorithms. These help detect errors in data entry and transmission.
Frequently Asked Questions
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