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Ireland National Math Olympiad
2009 Irish Math Olympiad
2
GCD of Products
GCD of Products
Source: 2009 IrMO Paper 2 Problem 2
January 30, 2018
number theory
greatest common divisor
Problem Statement
For any positive integer
n
n
n
define
E
(
n
)
=
n
(
n
+
1
)
(
2
n
+
1
)
(
3
n
+
1
)
⋯
(
10
n
+
1
)
.
E(n)=n(n+1)(2n+1)(3n+1)\cdots (10n+1).
E
(
n
)
=
n
(
n
+
1
)
(
2
n
+
1
)
(
3
n
+
1
)
⋯
(
10
n
+
1
)
.
Find the greatest common divisor of
E
(
1
)
,
E
(
2
)
,
E
(
3
)
,
…
,
E
(
2009
)
.
E(1),E(2),E(3),\dots ,E(2009).
E
(
1
)
,
E
(
2
)
,
E
(
3
)
,
…
,
E
(
2009
)
.
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