Number minus Modulo is Integer Multiple

From ProofWiki
Jump to navigation Jump to search

Theorem

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

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

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

where $\floor {\dfrac x y}$ denotes the floor of $\dfrac x y$.


Let $y < 0$.


Then:

$x - \paren {x \bmod y}$

is an integer multiple of $y$.


Proof

When $y = 0$ we have:

$x \bmod y := x$

Thus:

$x - \paren {x \bmod y} = 0$

From Zero is Integer Multiple of Zero it follows that:

$x - \paren {x \bmod y}$

is an integer multiple of $y$.


Let $y \ne 0$.

Then:

\(\ds x \bmod y\) \(=\) \(\ds x - y \floor {\dfrac x y}\) Definition of Modulo Operation
\(\ds \leadsto \ \ \) \(\ds x - \paren {x \bmod y}\) \(=\) \(\ds y \floor {\dfrac x y}\) Definition of Modulo Operation

From Floor Function is Integer, $\floor {\dfrac x y}$ is an integer.

Thus:

$\exists n \in \Z: x - \paren {x \bmod y} = n y$

where in the case $n = \floor {\dfrac x y}$

Hence the result by definition of integer multiple.

$\blacksquare$


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Number_minus_Modulo_is_Integer_Multiple&oldid=540229"