NL-NWO 2000 Problem 1

Let $a$ and $b$ be integers. Define $a$ to be a power of $b$ if there exists a positive integer $n$ such that $a = b^n$ . Define $a$ to be a multiple of $b$ if there exists an integer $n$ such that $a = bn$ .

Let $x$ , $y$ and $z$ be positive integer such that $z$ is a power of both $x$ and $y$ .

Decide for each of the following statements whether it is true or false. Prove your answers.

(a) The number $x + y$ is even. (b) One of $x$ and $y$ is a multiple of the other one. (c) One of $x$ and $y$ is a power of the other one. (d) There exist an integer $v$ such that both $x$ and $y$ are powers of $v$ (e) For each power of $x$ and for each power of $y$ , an integer $w$ can be found such that $w$ is a power of each of these powers. (f) There exists a positive integer $k$ such that $x^k > y$ .