How does division produce a quotient and sometimes a remainder?

In arithmetic, division is the process of splitting a dividend by a divisor to determine how many times the divisor is contained within the dividend. This relationship is expressed by the equation $a=q\times b+r$, where $a$ is the dividend, $b$ is the divisor, $q$ is the quotient, and $r$ is the remainder.12

The quotient represents the number of times the divisor fits into the dividend. If the dividend is an exact multiple of the divisor, the remainder is zero, and the quotient is the exact result of the division. However, if the dividend is not an exact multiple, there will be a remainder $r$ that is less than the divisor $b$. This remainder accounts for the portion of the dividend that cannot be evenly divided by the divisor. This fundamental relationship is central to procedures such as the Euclidean algorithm, which uses successive division and remainders to find the greatest common divisor of two numbers.123