January+26

9.1 a) Find a number math 0\leq a < 73 math with math a\equiv 9^{794} \mod 73 math (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 math 7^{1734250}\equiv 1660565 \mod 1734251 math holds, show that 1734251 is a composite (i.e. non-prime) number.

c) The congruence math 2^{52632}\equiv 1 \mod 52633 math is true. Can you conclude that 52633 is a prime number?

10.3 Show that for math m=561=3\cdot 11\cdot 17 math the congruence math a^{m-1}\equiv 1 \mod m math 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 math \phi(8800) math

11.5 Find x that solves math x\equiv 3 \mod 37 math and math x\equiv 1 \mod 87 math