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Problems
Contests
National and Regional Contests
Kazakhstan Contests
Kazakhstan National Olympiad
2015 Kazakhstan National Olympiad
4
4
Part of
2015 Kazakhstan National Olympiad
Problems
(1)
perfect square
Source:
1/10/2015
P
k
(
n
)
P_k(n)
P
k
(
n
)
is the product of all positive divisors of
n
n
n
that are divisible by
k
k
k
(the empty product is equal to
1
1
1
). Show that
P
1
(
n
)
P
2
(
n
)
⋯
P
n
(
n
)
P_1(n)P_2(n)\cdots P_n(n)
P
1
(
n
)
P
2
(
n
)
⋯
P
n
(
n
)
is a perfect square, for any positive integer
n
n
n
.
number theory unsolved
number theory
Kazakhstan