Number theory- mobius,multiplicative,primitive root

1

i) When do we say that a function f:**N**-->**C** is multipicative?

ii)Define mobius funtion 'mu' and show it is multiplicative.

iii)Given F(n)=SUMf(d) for all nE**N**,where f:**N**-->**C**

....................d|n

and the sum os over all natural divisors d of n,how can we compute f(n) using F and 'mu'.

iv)Use the Euclidean algorithm to find a,bE**Z** such that

1=25a+81b

v) Transform the congruence

649x=85 mod 2025

into a system of congruences and then find all solution xE**Z** by applying the chinese remainder theorem.

2

i) for mEN ,define the mulitiplicative group mod m.how many elements does it contain?

ii)what is primitive root mod m?define primitive root and index modulo a prime number p.

iii)Show that 2 is primitive root mod 11 and compute the index ind2a (mod11) for all aE{1,2.....,10}

iv) use your index table from iii) to find all xE**Z** satisfying 5x^7=3 (mod11)

v)Let p=prime and let x be a solution of the congruence

x^k=b (mod p )

under which condition on p and k is the solution x unique modulo p?

Jutisfy your anser by relating it to a result about linear congruences.

vi)show that 2 is a primitive root mod13.

vii)define quatratic residue modulo a prime number p

whether 67 is a quadratic residue mod 103 or not?

Thank you so much, it would help alot for my following resit exams.