1. primitive roots for primes

(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?

2. Originally Posted by mndi1105
(a) find the number of incongruent roots modulo 6 of the polynomial f(x)=x^2 - x
You have, $x(x-1)\equiv 0(\bmod 6)$. Thus, $x=0,1,3,4$. Therefore, there are six solutions up to congruence.
(b) why does the behavior exhibited in part (a) not violate Lagrange's Theorem?
Because that theorem applies only for $f(x) \equiv 0 (\bmod p)$ where $p$ is a prime.

3. How did you get that there were 6 incongruent roots modulo 6?

4. I think that was a mistake.

5. Originally Posted by mndi1105
How did you get that there were 6 incongruent roots modulo 6?
Originally Posted by whipflip15
I think that was a mistake.
Of that is is a mistake! If you look above we get four solutions if you count them, but I wrote six.