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2021 Thailand Online MO
P9
P9
Part of
2021 Thailand Online MO
Problems
(1)
τ(τ(an)) = τ(τ(bn)) for all n implies a=b
Source: 2021 Thailand Online MO P9 (Mock TMO contest)
4/6/2021
For each positive integer
k
k
k
, denote by
τ
(
k
)
\tau(k)
τ
(
k
)
the number of all positive divisors of
k
k
k
, including
1
1
1
and
k
k
k
. Let
a
a
a
and
b
b
b
be positive integers such that
τ
(
τ
(
a
n
)
)
=
τ
(
τ
(
b
n
)
)
\tau(\tau(an)) = \tau(\tau(bn))
τ
(
τ
(
an
))
=
τ
(
τ
(
bn
))
for all positive integers
n
n
n
. Prove that
a
=
b
a=b
a
=
b
.
number theory