If p is a prime then for any integer a we have

a^p = a modulo p.

i.e.  If p is a prime and n is an integer then n^p–n is divisible by p.

Example : 7 is a prime so n ⁷ – n is divisible by 7 .
For n = 2 : 2 7 – 2 = 128 – 2 = 126 is divisible by 7

Questions
Q1 : what is the reminder when 11³⁹ is divided by 19
Q2. Find the reminder when 5⁹¹ is divided by 91
Q3. Find the following reminders
a. ₇₅75⁷⁵ is divided by 37
b. 2¹⁰⁰ is divided by 101
c. ₂₀ 51 ⁹⁷ is divided by 17

Corollary :
 n^q – n is divisible by q where q is a prime number or product of two prime numbers.

Important Points:
1. If P be a prime number such that a^p – b^p is divisible by p, then it is also divisible by p²

2. If an integer n is greater than 2, then the equation aⁿ + bⁿ = cⁿ has no solutions in non-zero integers a, b, and c. (Fermat’s Last Theorem)

3. If p be prime and a is prime to p, then a^(p-1) – 1 is multiple of p.