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12
A 12
A 12
Source:
May 25, 2007
Divisibility Theory
Problem Statement
Let
k
,
m
,
k,m,
k
,
m
,
and
n
n
n
be natural numbers such that
m
+
k
+
1
m+k+1
m
+
k
+
1
is a prime greater than
n
+
1
n+1
n
+
1
. Let
c
s
=
s
(
s
+
1
)
.
c_{s}=s(s+1).
c
s
=
s
(
s
+
1
)
.
Prove that the product
(
c
m
+
1
−
c
k
)
(
c
m
+
2
−
c
k
)
⋯
(
c
m
+
n
−
c
k
)
(c_{m+1}-c_{k})(c_{m+2}-c_{k})\cdots (c_{m+n}-c_{k})
(
c
m
+
1
−
c
k
)
(
c
m
+
2
−
c
k
)
⋯
(
c
m
+
n
−
c
k
)
is divisible by the product
c
1
c
2
⋯
c
n
c_{1}c_{2}\cdots c_{n}
c
1
c
2
⋯
c
n
.
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