Primes? Squares? Polynomials? Too easy!

Note that the polynomial given can be expressed as:

$(x - 20)(x - 503)(x - 509)$ .

Hence, the largest root of this polynomial is $509$ , which is a prime. Now, we have to find the value of $509^{2} (mod 12)$ .

First, we note that $509 \cong 1 or 3 (mod 4)$ and that $509 \cong 1 or 2 (mod 3)$ . Solving this system of linear congruences will easily yield that $509^2 \cong 1 (mod 12)$ and hence, the remainder when $509^{2}$ is divided by $12$ is 1.

To determine $\sqrt[3]{1}$ , we solve the polynomial $x^3 = 1$ , i.e. $x^3 - 1 = 0$ . Factoring yields: $x - 1)(x^2 + x + 1) = 0$ , i.e. $x = 1$ or $x = (-1 + \sqrt{3}i)/2$ or $x = (-1 - \sqrt{3}i)/2$ . This yields $a = 1, b = 0, c = -1/2, d = \sqrt{3}/2, e = -1/2, f = -\sqrt{3}/2$ and hence, $a + b + c + d + e + f$ = $\boxed{0}$ .