MathDB

Multiplicativity+infinity=identity

Source: 2021 Mexico Center Zone Regional Olympiad, problem 6

January 17, 2022

Mexicoalgebranumber theoryfunctional equationSequence

Problem Statement

The sequence

a_1,a_2,\dots

of positive integers obeys the following two conditions:
[*] For all positive integers

m,n

, it happens that

a_m\cdot a_n=a_{mn}

[*] There exist infinite positive integers

n

such that

(a_1,a_2,\dots,a_n)

is a permutation of

(1,2,\dots,n)

Prove that

a_n=n

for all positive integers

n

.
Proposed by José Alejandro Reyes González