This is not a correct way to ask for help. Nobody will do your homework for you! You should ask one question at a time, explain what you don't understand and show some kind of attempt.
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 nEN,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,bEZ such that
1=25a+81b
v) Transform the congruence
649x=85 mod 2025
into a system of congruences and then find all solution xEZ 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 xEZ 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.