Question

If x is a positive integer less than 100, then the number of x which make $\sqrt{1+2+3+4+x}$ an integer is

• A. 6
• B. 7
• C. 8
• D. 9

Answer 1

The correct option is B 7

Let, $\sqrt{1+2+3+4+5+x}=l$, where $l$ is an positive integer.
$\Rightarrow \sqrt{10+x}=l$
$\Rightarrow 10+x=l^2$
$\Rightarrow x=l^2-10$
Given, $1<x<100$
$\Rightarrow 1<l^2-10<100$
$\Rightarrow 11<l^2<110$
$\Rightarrow \sqrt{11}<l<\sqrt{110}$
$\Rightarrow 3.32<l<10.49$
$\Rightarrow l=4,5,6,7,8,9,10$ ($\because l\text{is an integer.}$)
Therefore, there are 7 values of $x$.