Problems on remainders p-1

Fermat’s Theorem:

·                     If p is a prime and a is any number prime to p, then a p-1 -1 is divisible by p.

Corollary 1:

·                     If p is a prime and a is any number whatever, then ap-a is divisible by p.

·                       The fifth power of any number N has the same right-hand digit as N.

Find remainders when ;

(1) 13 ²³⁹ divided by 16.              (2) 13 ⁴⁰ divided by 15

(3) 19²²⁰ divided by 23               ( 4) 39²² divided by 7

(5) 2¹⁰⁰³ divided by 25                 (6) 4⁹⁶ divided by 6