Question

Let $\left[K\right]$ denotes the greatest integer less than or equal to $K$. If number of positive integral solutions of the equation $\left[\frac{x}{\left[{\pi }^2\right]}\right]=\left[\frac{x}{\left[11\frac{1}{2}\right]}\right]$ is $n$, then the value of $n$ is :

Answer 1

$\left[\frac{x}{9}\right]=\left[\frac{x}{\left[11.5\right]}\right](\because {\pi }^2\in (9,10\left)\right)$
$\Rightarrow \left[\frac{x}{9}\right]=\left[\frac{x}{11}\right]$
Case $:1.\text{If}0\le \frac{x}{9}<1$ and $0\le \frac{x}{11}<1$
$\Rightarrow 0\le x<9$ and $0\le x<11$
So,positive integral solutions of $x$ are $\{1,2,3,\dots 8\}$

Case $:2.\text{If}1\le \frac{x}{9}<2$ and $1\le \frac{x}{11}<2$
$\Rightarrow 9\le x<18$ and $11\le x<22$
So,positive integral solutions of $x$ are $\{11,12,13,\dots 17\}$

Case $:3.\text{If}2\le \frac{x}{9}<3$ and $2\le \frac{x}{11}<3$
$\Rightarrow 18\le x<27$ and $22\le x<33$
So,positive integral solutions of $x$ are $\{22,23,24,\dots 26\}$

Case $:4.\text{If}3\le \frac{x}{9}<4$ and $3\le \frac{x}{11}<4$
$\Rightarrow 27\le x<36$ and $33\le x<44$
So,positive integral solutions of $x$ are $\{33,34,35\}$

Case $:5.\text{If}4\le \frac{x}{9}<5$ and $4\le \frac{x}{11}<5$
$\Rightarrow 36\le x<45$ and $44\le x<55$
So,positive integral solutions of $x$ are $\left\{44\right\}$

Case $:6.\text{If}5\le \frac{x}{9}<6$ and $5\le \frac{x}{11}<6$
$\Rightarrow 45\le x<54$ and $55\le x<66$
So,no positive integral solutions of $x$ is possible

$\therefore$Total number positive integral solutions for $x$ $=8+7+5+3+1=24$