Functions struggling with the modulo

Let $f(x) = \alpha x^3 + \beta x^2 + \gamma x + \delta$ , where $\alpha$ , $\beta$ , $\gamma$ and $\delta$ are integers. Then, we have:

$\begin{aligned} f(a) - f(b) & = \alpha (a^3-b^3) + \beta (a^2 - b^2) + \gamma (a-b) \\ & = (a-b)(\alpha (a^2 + ab + b^2) + \beta (a + b) + \gamma) \end{aligned}$

Since $a \equiv b \text{ (mod 9)}$ , we can write $a=9k+b$ , where $k$ is a natural number. Then, we have:

$\begin{aligned} \frac {|f(a) - f(b)|}{18} & = \frac {|(9k+b-b)(\alpha (a^2 + ab + b^2) + \beta (a + b) + \gamma)|}{18} \\ & = \frac {|9k(\alpha (a^2 + ab + b^2) + \beta (a + b) + \gamma)|}{18} \\ & = \left| \frac k2 (\alpha (a^2 + ab + b^2) + \beta (a + b) + \gamma) \right| \end{aligned}$

The decimal expansion of a positive integer $n$ divided by 2 is either 0 when $n$ is even or 5 when $n$ is odd, therefore, it is $\boxed{\text{terminating}}$ .