# Quotient of Modulo Operation with Modulus

## Theorem

Let $x, y \in \R$ be real numbers.

Let $x \bmod y$ denote the modulo operation:

$x \bmod y := \begin{cases} x - y \left \lfloor {\dfrac x y}\right \rfloor & : y \ne 0 \\ x & : y = 0 \end{cases}$

where $\left \lfloor {\dfrac x y}\right \rfloor$ denotes the floor of $\dfrac x y$.

Let $y \ne 0$.

Then:

$0 \le \dfrac x y - \left \lfloor {\dfrac x y}\right \rfloor = \dfrac {x \bmod y} y < 1$

## Proof

$\dfrac x y - \left \lfloor {\dfrac x y}\right \rfloor \in \left[{0 \,.\,.\, 1}\right)$

Thus by definition of half-open real interval:

$0 \le \dfrac x y - \left \lfloor {\dfrac x y}\right \rfloor < 1$

Then:

 $\displaystyle x \bmod y$ $=$ $\displaystyle x - y \left \lfloor {\frac x y}\right \rfloor$ Definition of Modulo Operation $\displaystyle \leadsto \ \$ $\displaystyle \frac {x \bmod y} y$ $=$ $\displaystyle \frac {x - y \left \lfloor {\dfrac x y}\right \rfloor} y$ $\displaystyle$ $=$ $\displaystyle \frac x y - \frac y y \left \lfloor {\dfrac x y}\right \rfloor$ $\displaystyle$ $=$ $\displaystyle \frac x y - \left \lfloor {\dfrac x y}\right \rfloor$

Hence the result.

$\blacksquare$