Question

the remainder when 2^2005 is divided by 9 is​

Answer 1

Given:

2²⁰⁰⁵ divided by 9

To find:

The remainder

Solution:

As per the remainder theorem, the dividend is equal to the sum of the product of divisor and quotient and remainder.

So if a number is divisible by the factor of it then we get the remainder as 0.

From the theorem we have

$\frac{(a-1)^{n}}{a} = 1$ if the value of the n is even

$\frac{(a-1)^{n}}{a} = -1$ if the value of the n is odd

here we have given the number as:

so the given number can be expressed as:

$\frac{(8)^{668}.2^{1}}{9}$

$\frac{(9-1)^{668}.2^{1}}{9}$ {668 is even}

$\frac{2^{1}}{9}$

Here the numerator is less than the denominator of the fraction so the remainder will be 2

The remainder 2²⁰⁰⁵ divided by 9 is 2