I am trying to find the remainder when $4^{96}$ is divided by 6.
SO using the cyclicity method,
Dividing $4^1$ by 6 gives remainder 4.
Dividing $4^2$ by 6 gives remainder 4.
Dividing $4^3$ by 6 gives remainder 4.
Dividing $4^4$ by 6 gives remainder 4.
Dividing $4^5$ by 6 gives remainder 4.
..
..
So by this method we get the answer as 4
.
But when we reduces this to
$ 4^{100} /6 $ to $ 2^{200} /6 $
which is equal to
$ 2^{199} /3 $
Then by using the cyclicity method:
Dividing $2^1$ by 3 gives remainder 2.
Dividing $2^2$ by 3 gives remainder 1.
Dividing $2^3$ by 3 gives remainder 2.
Dividing $2^4$ by 3 gives remainder 1.
Dividing $2^5$ by 3 gives remainder 2.
..
..
So accrding to this ,We get the answer as 2.
So which one is correct?
And how we are getting two different answers for the same numbers?
Thanks in advance.