Show that, for any x_{1}, x_{2}, x_{3}, \ldots, x_{k} and any y, we can find a polynomial f of degree k-1 with integer coefficients so that f\left(x_{1}\right) \equiv y \bmod q and f\left(x_{i}\right) \equiv 0 \bmod q for 2 \leq i \leq k.
Show that, for any x_{1}, x_{2}, x_{3}, \ldots, x_{k} and any y, we can find a polynomial f of degree k-1 with integer coefficients so that f\left(x_{1}\right) \equiv y \bmod q and f\left(x_{i}\right) \equiv 0 \bmod q for 2 \leq i \leq k.