MathNet 2016-part-02 — 184 problems (EN–VI) Question 133

A positive integer $n$ is called a Mozartian number if the numbers $1,2, \ldots, n$ together contain an even number of each digit (in base 10).
Prove:
(a) All Mozartian numbers are even.
(b) There are infinitely many Mozartian numbers.