Modaritmetikk

Modaritmetikk, also known as modular arithmetic, is a system of arithmetic for integers, where numbers "wrap around" after reaching a certain value, the modulus. This concept is fundamental in number theory and has applications in various fields, including computer science, cryptography, and error-correcting codes.

In modular arithmetic, two numbers are considered equivalent if they differ by a multiple of the modulus. This equivalence is denoted by the congruence relation, written as "a ≡ b (mod n)", where "a" and "b" are integers, and "n" is the modulus. For example, in the modulus 5, the numbers 7 and 2 are congruent because 7 - 2 = 5, which is a multiple of 5.

The basic operations of addition, subtraction, and multiplication can be performed in modular arithmetic. For instance, in modulus 5, (3 + 4) ≡ 2 (mod 5) because 3 + 4 = 7, and 7 ≡ 2 (mod 5). Similarly, (3 × 4) ≡ 2 (mod 5) because 3 × 4 = 12, and 12 ≡ 2 (mod 5).

Modular arithmetic is particularly useful in situations where only the remainder of a division is of interest. It simplifies calculations and provides a framework for understanding patterns and cycles in numbers. For example, in the Gregorian calendar, the day of the week repeats every 7 days, which can be modeled using modular arithmetic with a modulus of 7.