Integer division that rounds up

Introduction

Usual integer division rounds down: ${\frac {a}{b}}=\left\lfloor {\frac {a}{b}}\right\rfloor$ for $a,b\in \mathbb {N} ,b\neq 0$. To round up (if overflow is not an issue), you can use following algorithm with the usual roundig down division: $q={\frac {x+y-1}{y}}=\left\lceil {\frac {x}{y}}\right\rceil$

Proof

Proof is in two parts; 1st if $y$ divides $x$, and if not. Note that usual integer division rounds down.

Part 1. If $y$ divides $x$ we have $x=ay$ for some $a\in \mathbb {N} _{+}$. Thus we have ${\begin{aligned}\left\lceil {\frac {x+y-1}{y}}\right\rceil =\left\lceil {\frac {x}{y}}+{\frac {y-1}{y}}\right\rceil \\=\left\lceil {\frac {ay}{y}}+{\frac {y-1}{y}}\right\rceil \\={\frac {ay}{y}}\\={\frac {x}{y}}\end{aligned}}$ because $0\leq {\frac {y-1}{y}}<1$. This is ok.