For every prime number p\(\ge\) difine F(p) = \(\sum\limits^{\dfrac{p-1}{2}}_{k=1}k^{120},f\left(p\right)=\dfrac{1}{2}-\left\{\dfrac{F\left(p\right)}{p}\right\}\), where {X} = X -\(\left\{X\right\}\) is the fractional part of x. Find the value of  f(p).

Let p and q  be prime numbers with q>p>2 .Prove that \(2^{pq}-1\) has at least three distinct prime factors.

Find al positive integral solutions to \(p^a-1=2^n\left(p-1\right)\),where p is a prime.

A 10-digit nukber is called a cute number if its digits belong to the set {1,2,3}  and the difference of every pair consecutive digits is 1.

a, Find the total number of cute numbers.

b, Prove that the sum of all cute numbers is divisible by 1408.

Let A be the sum of the decimal digits of \(^{4444^{4444}}\)and B be the sum of the decimal digits of A. Find the sum of the decimal digits of B.

A square room has its floor paved with square tiles of the same size ( where only whole tiles are used and no tile is cut ) . There are two colors of tiles : white and blue . The blue tiles are used on the two diagonals of the floor . The rest of the floor is paved with white tiles . Find the number of tiles used for the whole floor , given that the number of white tiles is 839 more than the number of  blue tiles.

Fractions of the form \(\dfrac{1}{n}\), where n is an integer, are known as unit fractions.

1) Express the number 1 as a sum of six different unit fractions, given that four of them are \(\dfrac{1}{2},\dfrac{1}{9},\dfrac{1}{10},\dfrac{1}{18}\).

2) Express the number 1 as a sum of seven different unit fractions, given that five of them are \(\dfrac{1}{3},\dfrac{1}{5},\dfrac{1}{9},\dfrac{1}{15},\dfrac{1}{45}\).

\(a@b=ab-2\)

Calculate :

\(\left(...\left(\left(1@2\right)@3\right)@4...@2015\right)\)