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

12

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.

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App