Question

The number of positive integers n in the set $1,2,3,....,100$ for which the number $\frac{1^2+2^2+3^3+......+n^2}{1+2+3+.....+n}$ is an integer is

• A. $33$
• B. $34$
• C. $50$
• D. $100$

Answer 1

Answer: (B)

$34$

$\frac{n(n+1)(2n+1)}{6n(n+1)}\times \Rightarrow \frac{2n+1}{3}=k\Rightarrow n=\frac{3k-1}{2}$
$1\le \frac{3k-1}{2}\le 100\Rightarrow 3\le 3k\le 201\Rightarrow 1\le k\le \frac{201}{3}$
$1\le k\le 67$
$\therefore$ Number of odd integers $=34$