Show that if:

F(xy) = F(x)F(y);

F(30) = 1;

F(10a+7) = 1 for any integer a 0< a

Then there is either simply one solution, F(x) = 1, or there are more functional equations that fit the above prerequisites. Provide justification.

I understand this is a surjective function and F(30/x) = F(7/x) = F(17/x) etc but I dont know how to identify any more solutions. I have been 'fiddling' for a while and my lack of functional equation knowledge is showing.