<p>Find the number of positive integers <img alt="$n$" height="8" src="https://latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png" width="10" /> less than <img alt="$1000$" height="13" src="https://latex.artofproblemsolving.com/6/e/e/6ee927e1332358c96c62c277441c907c4f51057f.png" width="35" /> for which there exists a positive real number <img alt="$x$" height="10" src="https://latex.artofproblemsolving.com/2/6/e/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png" width="12" /> such that <img alt="$n=x\lfloor x \rfloor$" height="19" src="https://latex.artofproblemsolving.com/f/e/d/fed03d4045f13580853bc7c804d90838ddbdde71.png" width="69" />.</p> <p>Note: <img alt="$\lfloor x \rfloor$" height="19" src="https://latex.artofproblemsolving.com/0/6/6/06681f5b345750104366c512078960fee8d81b49.png" width="23" /> is the greatest integer less than or equal to <img alt="$x$" height="10" src="https://latex.artofproblemsolving.com/2/6/e/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png" width="12" />.</p>

Question No: 243

Find the number of positive integers $n$ less than $1000$ for which there exists a positive real number $x$ such that $n=x\lfloor x \rfloor$ .

Note: $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$ .

Editorial

Tags: Algebra

Question Details