Question

If $n$ is the smallest natural number such that $n+2n+3n+\cdots +99n$ 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 $3$

$n+2n+3n+\cdots +99n$
$=n(1+2+3+\cdots +99)$
$=n\times \frac{99}{2}\times 100$
$=n\times 99\times 50$
$=n\times 11\times 3^2\times 5^2\times 2$
$=3^2\times 5^2\times 11\times 2\times n$
So, the smallest required number is $n=11\times 2=22$
$n^2=484$