QUESTION FOR SSC , UPSC , CDS AND ALL GENERAL COMPETITIONS

Find the remainder when (3^27) is divided by 5?

Firstly we will separate 3^27 like this, 

             (3^26).3  (it means 3^27 only) 

Then,

We can also write 3^26 as ((3^2)^13) 

We just splitted the power of 3

Now, 

          {((3^2)^13).3}/5.......... (i) 

If we divide 3^2 i.e. 9 by 5 remainder comes to be 4

And that means 4^13 and 4 is less than 5 so remainder again comes to be ( -1) and (-1)^13 

means 

-1 and in eq (i) 3 is also there left to be divided by 5 and hence there is no power on 3 so 

remainder will also be 3 

Atlast we have {(-1)*3}/5 

-3/5 means 5-3=2 (if there is - sign in the numerator then we have to substract the numerator 

from denominator only if numerator is smaller than denominator)

So, 

   2 will be the remainder of this question. 

FIND THE REMAINDER WHEN 3^1000 IS DIVIDED BY 7

3^1000 = 3^4 * 3^996

3^996 is divided by 7 and gives the remainder 1 

by, 

996 = 6k 

where 6 is the eulers's number for 7

Thus remainder when 3^4 is divided by 7, which is 4

or 

3^3 when divided by 7, the remainder comes to be -1

3^1000 = [3^3]^333 * 3

[-1]^333*3 = -1*3 = -3

then, the answer will be 7-3 = 4

YOU CAN ALSO LEARN SUCH REMAINDERS LIKE WHEN 3^N IS DIVIDED BY 7

3^995 = 5

3^996 = 1

3^997 = 3

3^998 = 2

3^999 = 6

3^1000 = 4