Question

The number of positive integral solutions of the equation is $\left[\frac{x}{99}\right]=\left[\frac{x}{101}\right]$ is$?$

• A. $2500$
• B. $2499$
• C. $1729$
• D. $1440$

Answer 1

The correct option is B

$2499$

Explanation for the correct option.

Find the number of solutions of the given equation.

An equation $\left[\frac{x}{99}\right]=\left[\frac{x}{101}\right]$ is given in the question.

Assume that, $\left[\frac{x}{99}\right]=\left[\frac{x}{101}\right]=p$.

So, for $p=0$.

The possible integral values of $x$ are $0,1,2,3,......,98$.

For $p=1$.

The possible integral values of $x$ are $101,102,103,......,197$.

For $p=2$.

The possible integral values of $x$ are $202,203,204,......,296$. and so on.

So, the total number of integral values of $x$ can be given by $\left[99+97+95+....1\right]-1$.

Find the sum $s$ of AP $\left[99+97+95+....1\right]$.

$$\begin{gathered} s=\frac{50}{2}[2·99+(50-1\left)\right(-2\left)\right] \\ \Rightarrow s=25[198-98] \\ \Rightarrow s=2500 \end{gathered}$$

Therefore, the total number of integral solutions of the given equation is $2499$.

Hence, option $B$ is the correct option.