Show that 12 1 is divisible by 13 by grouping together

Question

Exercise #1)  Show that 12!+ 1 is divisible by 13, by grouping together pairs of inverses modulo 13 thatoccur in 12!.

Exercise #2)  What is the remainder when 5!25! is divided by 31?

Exercise #3)  What is the remainder when 7 ×8 ×9 ×15 ×16 ×17×23 ×24 ×25 ×43 is divided by 11?

Exercise #4)  What is the remainder when 6²⁰⁰⁰ is divided by 11?

Exercise #5)  Using Fermat's little theorem, find the last digit of the base 7 expansion of 3¹⁰⁰.

Exercise #6)  Show that 30 | (n⁹- n) for all positive integers n.

Exercise #7)  If p is prime and k is a positive integer less than p,then the binomial coefficient(p_k)is divisible by p. Use this fact and the binomial theoremto show that if a and b are integers, then (a + b)^p ≡ a^p + b^p (mod p).