Kongruensregler

Kongruensregler, also known as modular arithmetic rules, are fundamental properties used in the study of congruences. A congruence is an equality relation between two integers that holds true modulo a third integer, called the modulus. The notation $a \equiv b \pmod{n}$ means that $a$ and $b$ have the same remainder when divided by $n$, or equivalently, that their difference $a-b$ is divisible by $n$.

Several key congruence rules simplify calculations and proofs. The reflexive property states that any integer is congruent to itself modulo $n$, i.e., $a \equiv a \pmod{n}$. The symmetric property asserts that if $a \equiv b \pmod{n}$, then $b \equiv a \pmod{n}$. The transitive property dictates that if $a \equiv b \pmod{n}$ and $b \equiv c \pmod{n}$, then $a \equiv c \pmod{n}$.

Further rules allow for arithmetic operations within congruences. If $a \equiv b \pmod{n}$ and $c \equiv d \pmod{n}$, then their sums are congruent: $a+c \equiv b+d \pmod{n}$. Similarly, their differences are congruent: $a-c \equiv b-d \pmod{n}$. The product of congruent numbers is also congruent: $ac \equiv bd \pmod{n}$.

These congruence rules are essential in various mathematical fields, including number theory, cryptography, and computer science, providing a framework for solving problems involving divisibility and remainders.