Integer division that rounds up

Introduction

Usual integer division rounds down: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac ab = \left \lfloor \frac ab \right \rfloor} for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a,b \in \mathbb N, b\neq 0} . To round up (if overflow is not an issue), you can use following algorithm with the usual roundig down division: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q = \frac{x+y-1}{y} = \left \lceil \frac xy \right \rceil }

Proof

Proof is in two parts; 1st if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} divides Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} , and if not. Note that usual integer division rounds down.

Part 1. If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} divides $x$ we have $x=ay$ for some $a\in \mathbb {N} _{+}$. Thus we have

${\begin{aligned}\left\lfloor {\frac {x+y-1}{y}}\right\rfloor &=\left\lfloor {\frac {x}{y}}+{\frac {y-1}{y}}\right\rfloor \\&=\left\lfloor {\frac {ay}{y}}+{\frac {y-1}{y}}\right\rfloor \\&={\frac {ay}{y}}\\&={\frac {x}{y}}\end{aligned}}$

because Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 \leq \frac{y-1}y < 1} . This part is ok.

Part 2. If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} does not divide Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=by + r} for some Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b\in\mathbb N_+} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0<r<b} . Thus we have

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \left \lfloor \frac{x+y-1}{y} \right \rfloor &= \left \lfloor \frac{x}{y} + \frac{y-1}y \right \rfloor \\ &= \left \lfloor \frac{by+r}{y} + \frac{y-1}y \right \rfloor \\ &= \left \lfloor \frac{by}{y} + \frac ry + \frac yy - \frac 1y \right \rfloor \\ &= \left \lfloor \frac{by + y}{y} + \frac {r-1}y \right \rfloor \\ &= \left \lfloor \frac{(b+1)y}{y} + \frac {r-1}y \right \rfloor \\ &= \frac{(b+1)y}{y} \\ \end{align} }

Combine the results, and we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left \lfloor \frac{x+y-1}{y} \right \rfloor = \left \lceil \frac{x}{y} \right \rceil } which was the question.