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Problems
Contests
National and Regional Contests
Mexico Contests
Mexico National Olympiad
2015 Mexico National Olympiad
6
6
Part of
2015 Mexico National Olympiad
Problems
(1)
Prove n is square-free
Source: Mexico National Olympiad 2015 Problem 6
11/25/2015
Let
n
n
n
be a positive integer and let
d
1
,
d
2
,
…
,
d
k
d_1, d_2, \dots, d_k
d
1
,
d
2
,
…
,
d
k
be its positive divisors. Consider the number
f
(
n
)
=
(
−
1
)
d
1
d
1
+
(
−
1
)
d
2
d
2
+
⋯
+
(
−
1
)
d
k
d
k
f(n) = (-1)^{d_1}d_1 + (-1)^{d_2}d_2 + \dots + (-1)^{d_k}d_k
f
(
n
)
=
(
−
1
)
d
1
d
1
+
(
−
1
)
d
2
d
2
+
⋯
+
(
−
1
)
d
k
d
k
Assume
f
(
n
)
f(n)
f
(
n
)
is a power of 2. Show if
m
m
m
is an integer greater than 1, then
m
2
m^2
m
2
does not divide
n
n
n
.
number theory