MathDB

For a positive integer, we define it's set of exponents the unordered list of all the exponents of the primes, in it`s decomposition. For example,

18=2\cdot 3^{2}

has it`s 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

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