Find the remainder when 65203 is divided by 7.

Answer: Option C

$$\begin{array}{rl}& \frac{{65}^{203}}{7}\\ & Or,\frac{{\left(63+2\right)}^{203}}{7}\\ & 63\text{is}\text{divisible}\text{by}7,\\ & \text{so}\text{remainder}\text{will}\text{depend}\text{on}\text{the}\text{powers}\text{of}2\\ & Or,\frac{2^{203}}{7}\\ & \text{Its}\text{remainder}\text{would}\text{be}\text{same}\text{as}\\ & \frac{2^3}{7}\\ & \text{Now,}\text{Required}\text{Remainder}\text{wold}\text{be},\\ & \frac{8}{7}=1\\ & \text{Required}\text{remainder}=1\end{array}$$
Note: We have manipulated the powers in the form of (4x + n). It means 203 is taken as,
203 = 4x + n = 4 × 50 + 3.
We neglect power which is in the multiple of 4.