MathDB
Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
2024 Vietnam National Olympiad
6
6
Part of
2024 Vietnam National Olympiad
Problems
(1)
We're back with arithmetic functions!
Source: 2024 Vietnam National Olympiad - Problem 6
1/6/2024
For each positive integer
n
n
n
, let
τ
(
n
)
\tau (n)
τ
(
n
)
be the number of positive divisors of
n
n
n
.a) Find all positive integers
n
n
n
such that
τ
(
n
)
+
2023
=
n
\tau(n)+2023=n
τ
(
n
)
+
2023
=
n
. b) Prove that there exist infinitely many positive integers
k
k
k
such that there are exactly two positive integers
n
n
n
satisfying
τ
(
k
n
)
+
2023
=
n
\tau(kn)+2023=n
τ
(
kn
)
+
2023
=
n
.
number theory