modulaariluvuissa

Modulaariluvuissa, commonly referred to as modular arithmetic, is a system of arithmetic for integers where numbers "wrap around" upon reaching a certain value—the modulus. This means that in modular arithmetic, we are only concerned with the remainders after division by a fixed integer, the modulus.

For example, in modulo 12 arithmetic, the numbers are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and 11. After 11 comes 0 again. So, 13 is equivalent to 1 in modulo 12, because 13 divided by 12 has a remainder of 1. Similarly, 25 is equivalent to 1 in modulo 12.

The congruence relation, denoted by the symbol $\equiv$, is fundamental to modular arithmetic. If two integers $a$ and $b$ have the same remainder when divided by a modulus $n$, then $a$ is congruent to $b$ modulo $n$, written as $a \equiv b \pmod{n}$. This is equivalent to saying that their difference $(a-b)$ is an integer multiple of $n$.

Modular arithmetic has widespread applications in various fields, including cryptography, computer science (for hashing functions and error correction codes), and number theory. It provides a powerful tool for simplifying complex calculations and for understanding the properties of integers. The set of integers modulo $n$, denoted as $\mathbb{Z}_n$ or $\text{Z}_n$, forms a ring, and under certain conditions, a field, which are important algebraic structures.