Product Distributes over Modulo Operation

Theorem

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

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

Then:

$z \paren {x \bmod y} = \paren {z x} \bmod \paren {z y}$

By definition of modulo operation:

$x \bmod y := \begin {cases} x - y \floor {\dfrac x y} & : y \ne 0 \\ x & : y = 0 \end {cases}$

If $y = 0$ we have that:

$z \paren {x \bmod 0} = z x = \paren {z x} \bmod \paren {z 0}$

If $y \ne 0$ we have that:

$\ds z \paren {x \bmod y}$

$=$

$\ds z x - z y \floor {\frac x y}$

$\ds $

$=$

$\ds z x - z y \floor {\frac {z x} {z y} }$

$\ds $

$=$

$\ds \paren {z x} \bmod \paren {z y}$

$\blacksquare$