Modular Arithmetic
(a op b) mod m operations
Calculates (a op b) mod m
About This Calculator
Modular arithmetic, sometimes called clock arithmetic, studies remainders after division. It is fundamental to computer science, cryptography (RSA encryption relies on it), hashing algorithms, and cyclic processes like days of the week.
Formula
(a + b) mod m = ((a mod m) + (b mod m)) mod m
(a x b) mod m = ((a mod m) x (b mod m)) mod m
Modular inverse: a^-1 mod m where a x a^-1 ≡ 1 (mod m)
Example Calculation
Compute (17 + 25) mod 7 and (17 x 25) mod 7
- (17 + 25) mod 7 = 42 mod 7 = 0
- 17 mod 7 = 3, 25 mod 7 = 4
- (3 + 4) mod 7 = 7 mod 7 = 0 (confirmed); (3 x 4) mod 7 = 12 mod 7 = 5
(17+25) mod 7 = 0; (17x25) mod 7 = 5
Addition mod 7 (partial table)
| + mod 7 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| 0 | 0 | 1 | 2 | 3 |
| 1 | 1 | 2 | 3 | 4 |
| 2 | 2 | 3 | 4 | 5 |
| 3 | 3 | 4 | 5 | 6 |
| 4 | 4 | 5 | 6 | 0 |
| 5 | 5 | 6 | 0 | 1 |
| 6 | 6 | 0 | 1 | 2 |
Frequently Asked Questions
What does a ≡ b (mod m) mean?
It means a and b leave the same remainder when divided by m, or equivalently that m divides (a - b). For example, 17 ≡ 3 (mod 7) because 17 - 3 = 14 is divisible by 7.
How is modular arithmetic used in cryptography?
RSA encryption relies on the fact that computing a^b mod n is easy, but reversing it (finding a from the result) is computationally infeasible for large n. This asymmetry secures internet communications.
What is a modular inverse?
The modular inverse of a (mod m) is a number x such that a*x ≡ 1 (mod m). It exists only when GCD(a, m) = 1. For example, the inverse of 3 mod 7 is 5, because 3*5 = 15 ≡ 1 (mod 7).