Understanding operations (addition, multiplication, inverse) in a modular system.
Modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus. A simple example is a 12-hour clock; if it's 9 o'clock, 4 hours later it will be 1 o'clock, not 13. This is arithmetic modulo 12. We write this as `(9 + 4) mod 12 = 1`. Formally, we say `a` is congruent to `b` modulo `m`, written `a ≡ b (mod m)`, if `m` divides `a - b`. This system is incredibly useful in programming, especially when dealing with problems that involve very large numbers. By taking the modulus at each step of a calculation, we can keep the intermediate results within the range of standard integer types, preventing overflow errors. Key properties include: `(a + b) mod m = ((a mod m) + (b mod m)) mod m` and `(a * b) mod m = ((a mod m) * (b mod m)) mod m`. Division is more complex and involves finding the modular multiplicative inverse. The modular inverse of `a` modulo `m` is an integer `x` such that `(a * x) mod m = 1`. An inverse exists only if `a` and `m` are coprime (their GCD is 1). The inverse can be found using the Extended Euclidean Algorithm. Modular arithmetic is the foundation for many algorithms in number theory, cryptography (like RSA), and hashing functions.