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&tau;(&tau;(an)) = &tau;(&tau;(bn)) for all n implies a=b

Source: 2021 Thailand Online MO P9 (Mock TMO contest)

April 6, 2021

number theory

Problem Statement

For each positive integer

k

, denote by

\tau(k)

the number of all positive divisors of

k

, including

1

and

k

. Let

a

and

b

be positive integers such that

\tau(\tau(an)) = \tau(\tau(bn))

for all positive integers

n

. Prove that

a=b