Question 98
For each positive integer $N$ with $2k$ digits, let $\text{odd}(N)$ be the $k$-digit number obtained by writing the digits of odd order of $N$ and $\text{even}(N)$ be the $k$-digit number obtained by writing the digits of even order of $N$. For example, $\text{odd}(249035) = 405$ and $\text{even}(249035) = 293$. Prove that there is no positive integer $N$ with $2k$ digits such that $N = \text{odd}(N) \cdot \text{even}(N)$.