Question

find the remainder when 2 to the power 31 is divided by 5

Answer 1

Hi friend, Here's the required answer:-

2^31= 2147483648.

So, 2^31÷5= 429496729.6

Hope this helps you...

Answer 2

2³¹=2³⁰(2)
    =(2²)¹⁵(2)
    =4¹⁵(2)
    =(5-1)¹⁵(2)
Now expanding (5-1)¹⁵ which is of form (x-y)ⁿ using binomial expansion 
    =(15C₀5¹⁵(-1)⁰+15C₁5¹⁴(-1)¹+...........15C₁₅(5)⁰(-1)¹⁵)(2)-----------EQUATON 1
Now we can observe that all terms of expansion (5-1)¹⁵ ae divisble by 5 except the last term 15C₁₅(-1)¹⁵
So,if we divide EQUATION 1 with 5 all terms will have remainder 0 except the last term which is   (-1)(2)=-3 but remainder for some reason in a binomial expansion cannot be negative so if we add and subtract 2 to -3 
-3-2+2=-5+2  now if we divide it with 5 then remainder will be 2.