Video solution:

Click to load, then click again to play

Written solution:

Suppose $x$ is a number that satisfies these conditions. We know that $100 \leq x \leq 999.$

The first statement implies that $$\begin{align*}x &\equiv 2 \mod 6\\&\equiv -4 \mod 6\end{align*}$$ This, in turn, implies that $$x+4 \equiv 0 \mod 6 .$$

Similarly, the second statement implies that $$\begin{align*}x &\equiv 5 \mod 9 \\&\equiv -4 \mod 9 \end{align*}$$ This, in turn, implies that $$x+4 \equiv 0 \mod 9 .$$

Finally, the third statement implies that $$\begin{align*}x &\equiv 7 \mod 11 \\&\equiv -4 \mod 11\end{align*}$$ This, in turn, implies that $$x+4 \equiv 0 \mod 11.$$

Together, these three conditions mean that $6\mid x+4,$ $9\mid x+4,$ and $11\mid x+4,$ so $$\operatorname{lcm}(6,9,11)\mid x+4 .$$ Therefore, $198\mid x+4.$ We also know $104 \leq x+4 \leq 1003 ,$ so we can see that there are $5$ possible values in this interval that are multiples of $198.$

Thus, E is the correct answer.