You just need one number to do it? I mean clearly 1 will always satisfy this requirement.

Less trivially, if , then certainly . This means a is invertible, so this is equivelent to seeing which elements have order dividing p-1 (mod ). because

(mod ).

Now one thing to consider is the order of this multiplicative group is . This means for every a that is relatively prime to , will satisfy this property.

For instance lets go mod

(mod 49) works and (mod 49)

(mod 49) works because (mod 49)

(mod 49) works because (mod 49)

You will need to skip things that are not relatively prime to like p, 2p, 3p, etc... I am not sure that this will generate everything, nor are these necessarily all going to be unique, but there are certainly quite a few numbers that should satisfy this property.