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IMO Shortlist
2016 IMO Shortlist
N2
N2
Part of
2016 IMO Shortlist
Problems
(1)
Integral ratio of divisors to divisors 1 mod 3 of 10n
Source: 2016 IMO Shortlist N2
7/19/2017
Let
τ
(
n
)
\tau(n)
τ
(
n
)
be the number of positive divisors of
n
n
n
. Let
τ
1
(
n
)
\tau_1(n)
τ
1
(
n
)
be the number of positive divisors of
n
n
n
which have remainders
1
1
1
when divided by
3
3
3
. Find all positive integral values of the fraction
τ
(
10
n
)
τ
1
(
10
n
)
\frac{\tau(10n)}{\tau_1(10n)}
τ
1
(
10
n
)
τ
(
10
n
)
.
number theory
IMO Shortlist