Question

If $0<x<1000$ and $\left[\frac{x}{2}\right]+\left[\frac{x}{3}\right]+\left[\frac{x}{5}\right]=\frac{31}{30}x$; where $\left[x\right]$ is the greatest integer less than or equal to $x$, the number of possible values of $x$ is

• A. $34$
• B. $32$
• C. $33$
• D. none of these

Answer 1

The correct option is C $33$

LHS$=$RHS iff $x$ is multiple of $2,3,5$
So $x$ should be a multiple of LCM of $2,3,5=30$
Total no of multiples between $0$ and $1000$ for $30$ is $33$