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The remainder when $2^{2000}$ is divided by 17 is

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answer is A.

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Detailed Solution

We have $2^{4} = 16 = 17 - 1$

$\begin{array}{l} ∴ 2^{2000} = {(2^{4})}^{500} = (17 - 1)^{500} \\ =^{500} C_{0} 17^{500} -^{500} C_{1} 17^{499} +^{500} C_{2} 17^{498} -^{500} C_{3} 17^{497} + \dots -^{500} C_{499} (17) + (- 1)^{500} \\ = 17 m + 1 \end{array}$

where m is some positive integer.
Thus, the required remainder is 1.

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