remainder

In mathematics, a remainder is what is left over after division. For example, when 7 is divided by 3, the quotient is 2, and the remainder is 1. This can be expressed as (7 = 2 \times 3 + 1). The remainder is always less than the divisor.

The concept of remainders is fundamental to modular arithmetic, where calculations are performed within a cycle of a specific number (the modulus). For instance, a 24-hour clock uses modular arithmetic with a modulus of 24, resetting to 0 at midnight.

The remainder theorem is a key concept in algebra. It states that when a polynomial (f(x)) is divided by (x - a), the remainder is (f(a)). The Euclidean algorithm is a procedure for efficiently finding the greatest common divisor of two numbers by repeatedly calculating remainders.