Determine all possible values of an integer N such that each of the roots of the equation 3X^{3} - 3X^{2} + N = 0 is a *rational number*.

Let F(x) = 3x^3 - 3x^2 + N

The first derivative of the polynomial is F'(x) = 9x^2 - 6x. Then F(x) has relative maximum at x=0 and relative minimum at x=2/3. Then 0<=N<=2/3 is necessary for F(x) to have three real roots.

The only integer in the range is N=0, which yields F(x) = 3x^3 - 3x^2 with rational roots {1,0,0}.