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Iran TST P2

Source: Iranian TST 2022 Problem 2

April 2, 2022

number theory

Problem Statement

For a positive integer

n

, let

\tau(n)

and

\sigma(n)

be the number of positive divisors of

n

and the sum of positive divisors of

n

, respectively. let

a

and

\tau(a)

be positive integers such that

\sigma(a^n)

divides

\sigma(b^n)

for all

n\in \mathbb{N}

. Prove that each prime factor of

\tau(a)

divides

\tau(b)

.
Proposed by MohammadAmin Sharifi

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