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1. Find the remainder when 2²⁵⁶ is divided by 17.

A. 1

B. 16

C. 14

D. None of these

Solution:

\begin{aligned} Given, \\ \frac{2^{256}}{17} \\ We can write it as : {(2^{4})}^{64} \\ \frac{16^{64}}{17} \\ Individually, when 16 is divided by 17, \\ gives a negative reminder of  - 1 . \\ Required Remainder, \\ {(- 1)}^{64} = 1 \\ Alternatively, \\ \frac{16^{64}}{17}, can be written as \\ \frac{(16 \times 16 \times 16 \times 16 \times 16 . . . . . . 64 times)}{17} \\ Now, we take the negative remainder of each, \\ 16 divided by 17 gives negative remainder - 1 . \\ So, the Remainder will be \\ (- 1 \times - 1 \times - 1 \times - 1 \times - 1 . . . . . . . 64 times) \\ = 1 \end{aligned}

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