Brilliant and Problem Solvers

The percentage of viewers who solved the problem, written as a decimal, can be written as $\frac{S}{V},$ where $S$ is the number of solvers and $V$ is the number of viewers. In this case, our fraction is $\frac{124}{V}.$ Thus, all integers $V$ such that $\frac{22}{100} \leq \frac{124}{V} < \frac{23}{100}$ will satisfy our problem. Solving this yields

$539\frac{3}{23} \leq V < 563\frac{7}{11}.$

Since $V$ is an integer, $540 \leq V \leq 563,$ and our sum is $540 + 541 + \cdots + 563 = 132\boxed{36}.$

(The actual number of viewers on the problem, as of 12/15/14, is 557.)

For exactly 22% we need (124*100)/22 viewers, which amounts to 563.63636363......

For exactly 23% we need (124*100)/23 viewers, which amounts to 539.13043478260869565217391304348.

Therefore, the required number of viewers should be between (and including) 540 - 563.

Therefore the required sum is 540+541+......+562+563=

whose last two digits are 36

Let $\alpha$ be the number of viewers.

Given when 124 people solved the questions, 22% got this problem right,so we can equate by standard method to calculate %

$(124*100)/\alpha \\$ = 22 $\Longrightarrow$

$\alpha$ = $(124*100)/22$ = 563.63636363636363636