The text defined the remainder function, rem, in terms of...

The text defined the remainder function, rem, in terms of the quotient function, quot. Use these definitions to show that
$$
\operatorname{rem}(n+d, d)=\operatorname{rem}(n, d) .
$$
Now, without referring to quot, write a recursive definition for the remainder function.

Key Concepts

Recursive Function Definition

A recursive definition specifies a function in terms of simpler cases of itself along with a base case. In the context of the remainder function, instead of using the quotient, one can define it such that if the integer is smaller than the divisor, the remainder is the integer itself (the base case), and if it is not, the function is defined recursively as the remainder of the integer minus the divisor. This mirrors the iterative process of subtracting the divisor until the number lies within the desired range.

Periodicity of the Remainder Function

The property expressed by rem(n + d, d) = rem(n, d) demonstrates that the remainder function is periodic with period equal to the divisor. This means that adding the divisor to a number does not change the remainder when divided by that divisor, a key notion in modular arithmetic that reflects the cyclical nature of remainders.

Division Algorithm

This fundamental theorem states that for any integer and a positive divisor, there exist unique integers (the quotient and the remainder) such that the original integer equals the product of the divisor and the quotient plus the remainder, where the remainder is nonnegative and smaller than the divisor. This concept underlies the method of computing the remainder and proves properties such as the periodicity observed when adding the divisor to the integer.

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