Given a positive integer \(n\) and a real number \(k\), consider the following equation in \(x\), \[(x-1)(x-2)(x-3) \times \cdots \times (x-n) = k.\] Which of the following statements about this equation is true?

If \(n=3\), then the equation has no real solution \(x\) for some values of \(k\).

If \(n\) is even, then the equation has a real solution \(x\) for any given value of \(k\).

If \(k \ge 0\) then the equation has (at least) one real solution \(x\).

The equation never has a repeated solution \(x\) for any given values of \(k\) and \(n\).