polynomial making with some features

Question

I need to find polynomial f(x) of degree 1000000, with integer coefficients such that:
1. It evaluates to an integer number in [0, 9] for all integers in [0, 1000000].
2. $f(x) \equiv f(1) \textrm{mod}\ (x - 1)^ 2$
Please note that is naturally satisfied that $f(x) \equiv f(1) \textrm{mod}\ x - 1$ because
$f(x) = (x - 1)Q(x) + f(1)$
where Q(x) is the quotient of dividing f(x) to (x - 1);
But we need something stronger here and it should satisfy the condition $f(x) \equiv f(1) \textrm{mod}\ (x - 1)^ 2$
Such a polynomial was a building block of a midterm exam question that assumed such f(x) exists and asked for something else. My curiosity brought me here to ask about the way to build such a polynomial. Can anyone help me with this?
Thanks!

Answer 1

$(1-x)(1-\frac {x}{2})(1-\frac {x}{3})\cdots (1-\frac{x}{1000000})$

Equals 0 for all integers in $[1,1000000]$ and equals 1 at $x= 0$