(a) find the number of incongruent roots modulo 6 of the polynomial f(x)=x^2 - x
(b) why does the behavior exhibited in part (a) not violate Lagrange's Theorem?
You have, $\displaystyle x(x-1)\equiv 0(\bmod 6)$. Thus, $\displaystyle x=0,1,3,4$. Therefore, there are six solutions up to congruence.
Because that theorem applies only for $\displaystyle f(x) \equiv 0 (\bmod p)$ where $\displaystyle p$ is a prime.(b) why does the behavior exhibited in part (a) not violate Lagrange's Theorem?