Question

Que: refer Picture :

Find the remainder when 2 power 2005 is divided By 9

HINT:
POWER FOLLOWS AP of which starting points amongst
7, 8, 9, 10, 11, 12 and refer Table in photo
example:
P = a +n6 where n = Natural Number. a is starting point from 7 to 12 . so ( Power - a from table) should be divisible By 6 . ie difference is Even and Sum of digits divisible by 3 .

Answer 1

Answer:

Hope it is correct if it is correct then mark as brainliest please☺☺

Answer 2

Answer:

The remainder when $2^{2005}$ is divided by $9$ is $2$.

Step-by-step explanation:

Consider the expression as follows:

${2^{2005}}$ ...... (1)

Compute the value of few powers of $2$ as follows:

$2^1=2$ cannot be divided by $9$

$2^2=4$ cannot be divided by $9$

$2^3=8$ cannot be divided by $9$

$2^4=16$ leaves remainder $7$ when divided by $9$

$2^5=32$ leaves remainder $5$ when divided by $9$

$2^6=64$

Notice that when $2^6$, i.e., $64$ is divided by $9$ leaves remainder $1$.

Rewrite expression (1) as follows:

⇒ $2^{2005}=(2^6)^{334} \cdot2^1$

Since the number $2^6$ leaves remainder $1$ when divided by $9$.

This implies the number $(2^6)^{334}$ also leaves remainder $1$ when divided by $9$.

So, the remainder when $(2^6)^{334} \cdot2^1$ is divided by $9$ is $2$.

Therefore, the remainder when $2^{2005}$ is divided by $9$ is $2$.

SPJ3