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National and Regional Contests
PEN Problems
PEN A Problems
58
58
Part of
PEN A Problems
Problems
(1)
A 58
Source:
5/25/2007
Let
k
≥
14
k\ge 14
k
≥
14
be an integer, and let
p
k
p_k
p
k
be the largest prime number which is strictly less than
k
k
k
. You may assume that
p
k
≥
3
k
4
p_k\ge \tfrac{3k}{4}
p
k
≥
4
3
k
. Let
n
n
n
be a composite integer. Prove that [*] if
n
=
2
p
k
n=2p_k
n
=
2
p
k
, then
n
n
n
does not divide
(
n
−
k
)
!
(n-k)!
(
n
−
k
)!
, [*] if
n
>
2
p
k
n>2p_k
n
>
2
p
k
, then
n
n
n
divides
(
n
−
k
)
!
(n-k)!
(
n
−
k
)!
.
Divisibility Theory