TUYMAADA 2006 Problem 8

For a positive integer, we define it's set of exponents the unordered list of all the exponents of the primes, in its decomposition. For example, $18=2\cdot 3^{2}$ has its set of exponents $1,2$ and $300=2^{2}\cdot 3\cdot 5^{2}$ has it`s set of exponents $1,2,2$ . There are given two arithmetical progressions $\big(a_{n}\big)_{n}$ and $\big(b_{n}\big)_{n}$ , such that for any positive integer $n$ , $a_{n}$ and $b_{n}$ have the same set of exponents. Prove that the progressions are proportional (that is, there is $k$ such that $a_{n}=kb_{n}$ for any $n$ ).

Proposed by A. Golovanov