Find the remainder when 83261 is divided by 17

Asked by admin @ 23/08/2021 in Math viewed by 329 People

Find the remainder when 83^261is divided by 17

Answered by admin @ 23/08/2021

Given:

83^261is divided by 17.

To find:

The remainder when 83^261is divided by 17.

Solution:

1) We will solve this by the remainder theorem.

2) We will represent 83 as the nearest multiple of the 17.

83 can be written as 85-2

so the remainder is given by:

$\frac{(85-2)^{261}}{17}$

3) 85 is the multiple of 17 so this will give you the remainder zero so the term will be:

$\frac{-2^{261}}{17}$

$\frac{-2^{260}.-2^{1}}{17}$

$\frac{{(-2^{4})}^{65}.-2^{1}}{17}$

$\frac{{16}^{65}.-2^{1}}{17}$

now 16 is represented as multiple of 17

$\frac{{(17-1)}^{65}.-2^{1}}{17}$

$\frac{{(-1)}^{65}.-2^{1}}{17}$

$\frac{{-1}.-2}{17}$

$\frac{2}{17}$

The numerator is less than the denominator so the remainder is the given numerator of the fraction only.

The remainder when 83^261is divided by 17 is 2

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