First slide

Binomial theorem for positive integral Index

Question

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

Moderate

A

1
B

2
C

8
D

none of these

Solution

$We have 2^{4} = 16 = 17 - 1$
$\begin{array}{l} ∴ 2^{2000} = {(2^{4})}^{500} = (17 - 1)^{500} \\ =^{500} C_{0} 17^{500} - {}^{500}C_{1} 17^{499} + {}^{500}C_{2} 17^{498} - {}^{500}C_{3} 17^{497} + \dots^{500} C_{499} (17) + (- 1)^{500} \end{array}$
=17m+1, where m is some positive integer.
$∴$ required remainder is 1.

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