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National and Regional Contests
PEN Problems
PEN A Problems
25
25
Part of
PEN A Problems
Problems
(1)
A 25
Source:
5/25/2007
Show that
(
2
n
n
)
∣
lcm
(
1
,
2
,
⋯
,
2
n
)
{2n \choose n} \; \vert \; \text{lcm}(1,2, \cdots, 2n)
(
n
2
n
)
∣
lcm
(
1
,
2
,
⋯
,
2
n
)
for all positive integers
n
n
n
.
number theory
least common multiple
floor function
logarithms
modular arithmetic
function
inequalities