Fermats little theorem on permutations

[lilcoley23@ho]

I'm looking at a card shuffle. And in shuffle would be the permutation (1, 2, 3, ..., n, n+1, n+2, n+3, ...2n) to (2, 4, 6, ..., 2n 1, 3, 5, ...2n-1) I know that it would take 52 perfect shuffles to get the deck of cards back in the original order. I think that I'm supposed to show this using Fermats Little Theorem.
Fermats little theorem says a ^p-1 ≡ 1(mod p) or a ^p ≡ a(mod p). Now I'm pretty sure that using the permutation from above plugging in for Fermat'l Little Theorem looks like: 52 ≡ 2(mod 2n+1)
If that is right I'm not sure how to show that I came up with that 52.
Can somebody please help?
Thanks you so much! Nicole

 

[tiny-tim]

… the permutation (1, 2, 3, ..., n, n+1, n+2, n+3, ...2n) to (2, 4, 6, ..., 2n 1, 3, 5, ...2n-1) …

Hi Nicole!

Hint: what is k/2 (mod 53)?