Abstract Albegra Question?

Question

How can I find all of the generators of order 9 in Z_36??

Answer 1

I suppose You mean all elements of order 9?
You must solve the congruence
9x ≡ 0 (mod 36), reducing both sides and the modulo by 9, that leads to x ≡ 0 (mod 4), so the elements are
4, 8, 12, 16, 20, 24, 28 and 32.
The multiples of each element of 9th order can have only 9 distinct values /listed above along with 0 in our case/ and cannot generate the entire group of 36.