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239 Open Math Olympiad
2004 239 Open Mathematical Olympiad
6
Divisibility of LCM
Divisibility of LCM
Source: 239 2004 J P6
May 15, 2020
number theory
least common multiple
Problem Statement
Given distinct positive integers
a
1
,
a
2
,
…
,
a
n
a_1,\,a_2,\,\dots,a_n
a
1
,
a
2
,
…
,
a
n
. Let
b
i
=
(
a
i
−
a
1
)
(
a
i
−
a
2
)
…
(
a
i
−
a
i
−
1
)
(
a
i
−
a
i
+
1
)
…
(
a
i
−
a
n
)
b_i = (a_i - a_1) (a_i-a_2) \dots (a_i-a_{i-1}) (a_i-a_{i+1})\dots(a_i-a_n)
b
i
=
(
a
i
−
a
1
)
(
a
i
−
a
2
)
…
(
a
i
−
a
i
−
1
)
(
a
i
−
a
i
+
1
)
…
(
a
i
−
a
n
)
. Prove that the least common multiple
[
b
1
,
b
2
,
…
,
b
n
]
[b_1,b_2,\dots,b_n]
[
b
1
,
b
2
,
…
,
b
n
]
is divisible by
(
n
−
1
)
!
.
(n-1)!.
(
n
−
1
)!
.
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