Question

If n is the smallest natural number such that n + 2n + 3n +....+ 99n is a perfect square, then the number of digits in n${}^2$ is

• A. 1
• B. 2
• C. 3
• D. More than 3

Answer 1

The correct option is C

3

n + 2n + 3n + $\dots$$\dots$$\dots$.. + 99n

n ((1 + 2 + 3 + $\dots$$\dots$ + 99)

$n\frac{\left(99\right)\left(100\right)}{2}$= 9 $\times$ 25 $\times$ 22 $\times$ n is a perfect square when n = 22

Number of digits in n = 3