Question 4.

There are multiple ways of solving such questions involving remainders; one easy way is to look for a power of numerator that leaves a remainder of 1 or -1 when divided by the denominator. 

In this instance, 1000, when divided by 13, leaves a remainder of -1

We can rewrite the numerator as $\frac{10^{66}\times\ 100}{13}$

The remainder would be $\left[\frac{10^{66}}{13}\right]_R\times\ \left[\frac{100}{13}\right]_R$

$\left(-1\right)^{22}\times\ 9$

9

Therefore, Option C is the correct answer.