Smallest n such that there's a polynomial in $\Bbb Z_n [X]$ of degree 4 that has 8 roots in $\Bbb Z_n$ 
I'm looking for the smallest positive integer $n$ such that there's a quartic polynomial in $\Bbb Z_n [X]$ that has 8 distinct roots in $\Bbb Z_n$.

I have n equal to 15 with roots 1, 2, 4, 7, 8, 11, 13, 14 and equation $x^4-1=0$ but I'm unsure if you can go lower (is it possible to have n equal to 8?) 
Also, while my math experience barely touches the surface of number theory and abstract algebra thus I'm not necessarily looking to prove this, I would like to understand why my claim of the minimality of n is true (or why yours for n less than 15 is) which I currently do not. 

Every number raied to $4$ ends in the same digit than the number itself, so $X^4-X$ has 10 roots in $\Bbb Z_{10}$, so $15$ is not the answer.
