The remainder when 30 + 31 + 32 + 33 + . . . . . . . + 3200 is divided by 13 is:

Answer: Option A

$$\begin{array}{rl}& \text{The}\text{given}\text{expression}\text{is}\text{in}\text{GP}\text{series}\\ & S=3^0+3^1+3^2+3^3+........+3^{200}\\ & S=\frac{3^0\times \left(3^{201}-1\right)}{3-1}\\ & S=\frac{3^{201}-1}{2}\\ & S=\frac{{\left(3^3\right)}^{67}-1^3}{2}\\ & S=\frac{{27}^{67}-1^3}{2}\end{array}$$
Since, (Aⁿ - Bⁿ) is divisible by (A - B), So, (27⁶⁷ - 1³) is divisible by (27 - 1) = 26
Hence, Expression is also divisible by 13 as it is divisible by 26
Thus given expression is divisible by 13 so the remainder will be 0