Modular Arithmetic Calculator
Calculate a mod n, modular addition, multiplication, and fast exponentiation a^b mod n. Essential for cryptography, hash functions, and number theory.
🔢 Number Theory📐 a mod n = remainder of a ÷ n🔢 Math
Operation
Value a
Modulus n
Value b
Please enter valid values.
Formula & Reference
| Variable | Symbol | Formula | Units |
|---|---|---|---|
| Modular Arithmetic Calculator | — | a mod n = remainder of a ÷ n | 0 to n-1 |
Step-by-Step Examples
Example 1
Simple
17 mod 5
- 17=3×5+2, remainder=2
✓ 2
Example 2
Multiplication
(7×8) mod 12
- 56=4×12+8, result=8
✓ 8
Example 3
Fast exponentiation
3^10 mod 7
- 3²≡2, 3⁴≡4, 3⁸≡2 (mod 7)
- 3^10=3^8·3^2≡2·2=4
✓ 4
Real-World Applications
Cryptography
RSA: c=m^e mod n.
Programming
Hash tables, circular buffers, checksums.
Clocks
12-hour clock: 9+5=14≡2 (mod 12).
Number Theory
Congruences, CRT, Euler's theorem.
Common Mistakes to Avoid
⚠️
Negative numbers
(-7) mod 3 = 2 mathematically. Use ((a%n)+n)%n to handle negatives.
⚠️
Don't compute a^b directly
For large b, use repeated squaring to avoid overflow.
⚠️
Mod vs remainder
Mathematical mod is always non-negative; computer % can be negative.
Frequently Asked Questions
What is modular arithmetic? ▾
Numbers wrap around at n. Like a clock face.
a≡b (mod n)? ▾
Same remainder when divided by n. n divides (a-b).
Modular inverse? ▾
x such that a·x≡1 (mod n). Used in RSA decryption.
What is RSA? ▾
Encrypt: m^e mod n. Decrypt: c^d mod n.
Chinese Remainder Theorem? ▾
Unique solution to system of congruences with coprime moduli.
Related Math Calculators
Formula Explorer connections
Interpretation: This formula uses integer divisibility, modular arithmetic, finite fields or coding relationships. Assumption: Inputs are usually integers with specific modulus or coprimality requirements. Cryptographic examples are educational and not a substitute for vetted security libraries.