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1. Find the remainder when 65²⁰³ is divided by 7.

A. 4

B. 2

C. 1

D. 6

Solution:

\begin{aligned} \frac{65^{203}}{7} \\ O r, \frac{{(63 + 2)}^{203}}{7} \\ 63 is divisible by 7, \\ so remainder will depend on the powers of 2 \\ O r, \frac{2^{203}}{7} \\ Its remainder would be same as \\ \frac{2^{3}}{7} \\ Now, Required Remainder wold be, \\ \frac{8}{7} = 1 \\ Required remainder = 1 \end{aligned}

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.

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