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Serbia JBMO TST
2022 Serbia JBMO TST
3
3
Part of
2022 Serbia JBMO TST
Problems
(1)
5 conditions
Source: Serbia JBMO TST 2022 P3
6/1/2022
Find all natural numbers
n
n
n
for which the following
5
5
5
conditions hold:
(
1
)
(1)
(
1
)
n
n
n
is not divisible by any perfect square bigger than
1
1
1
.
(
2
)
(2)
(
2
)
n
n
n
has exactly one prime divisor of the form
4
k
+
3
4k+3
4
k
+
3
,
k
∈
N
0
k\in \mathbb{N}_0
k
∈
N
0
.
(
3
)
(3)
(
3
)
Denote by
S
(
n
)
S(n)
S
(
n
)
the sum of digits of
n
n
n
and
d
(
n
)
d(n)
d
(
n
)
as the number of positive divisors of
n
n
n
. Then we have that
S
(
n
)
+
2
=
d
(
n
)
S(n)+2=d(n)
S
(
n
)
+
2
=
d
(
n
)
.
(
4
)
(4)
(
4
)
n
+
3
n+3
n
+
3
is a perfect square.
(
5
)
(5)
(
5
)
n
n
n
does not have a prime divisor which has
4
4
4
or more digits.
number theory