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Iran Contests
Iran Team Selection Test
2022 Iran Team Selection Test
2
2
Part of
2022 Iran Team Selection Test
Problems
(1)
Iran TST P2
Source: Iranian TST 2022 Problem 2
4/2/2022
For a positive integer
n
n
n
, let
τ
(
n
)
\tau(n)
τ
(
n
)
and
σ
(
n
)
\sigma(n)
σ
(
n
)
be the number of positive divisors of
n
n
n
and the sum of positive divisors of
n
n
n
, respectively. let
a
a
a
and
b
b
b
be positive integers such that
σ
(
a
n
)
\sigma(a^n)
σ
(
a
n
)
divides
σ
(
b
n
)
\sigma(b^n)
σ
(
b
n
)
for all
n
∈
N
n\in \mathbb{N}
n
∈
N
. Prove that each prime factor of
τ
(
a
)
\tau(a)
τ
(
a
)
divides
τ
(
b
)
\tau(b)
τ
(
b
)
.Proposed by MohammadAmin Sharifi
number theory