Express $X$ as a sum of two independent random variables

Question

We are given an integer $r \geq 2$ which is not a prime number and a random variable $X$ such that $P(X=k) = 1/r$ for $k= 0,1,...,r-1$, and $P(X=k) = 0$ otherwise. Express $X$ as a sum of two independent, non-constant random variables. I know I need to use probablity-generating functions somehow, but I am not sure how. How do I do that?

Answer 1

Since $r\ge2$ is not prime, write $r=ab$ with integers $a,\,b\ge2$. Then take $X=Y+Z$ with $Y\sim\mathcal{U}(0,\,a-1),\,Z/a\sim\mathcal{U}(0,\,b-1)$ independent. Equivalently, define $Z:=a\left\lfloor\frac{X}{a}\right\rfloor,\,Y:=X-Z$.