9.1 a) Find a number

with

(Hint: use Fermat's little theorem).

9.2 b) Show that for a prime number p the number (p-1)!+1 is divisible by p
(Hint: consider the number (p-1)! mod p. In this product one can split most of the residues to pairs of a residue and its inverse).

9.4 a) Assuming that the congruence

holds, show that 1734251 is a composite (i.e. non-prime) number.

c) The congruence

is true. Can you conclude that 52633 is a prime number?


10.3 Show that for

the congruence

holds for every a relatively prime to m.
(Hint: use Fermat's little theorem to compute a^{m-1} mod p for every prime divisor p of m).

11.1 b) Determine the value of


11.5 Find x that solves

and