After going through all the reminder chapters/notes on this site I made few points on calculating reminder. Here I am trying to put all those points together so that while approaching the reminder questions we can follow a certain pattern.

Question: what is the reminder when M is divided by N.

1. First Check if N is prime if so then use Fermat’s Little theorem.

2. If N is not prime but co-prime with M then use Euler’s Conecpt as given @

3. Use principle of cylicity.

4. Chinese Reminder Theorem is used when questions are asked like : a number when divided by a gives b and when divided by c gives d.

a. (any number x + 1) ^N when divided by x gives reminder 1.

b. (any number x ) ^N when divided by (x + 1) gives reminder 1 when N is even and reminder x when N is odd.

c. xⁿ + aⁿis exactly divisible by (x + a)
if n is odd, but not if n is even

d. xⁿ - aⁿ is divisible by (x + a)
if n is even but not if n is odd