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Caucasus Mathematical Olympiad
2021 Caucasus Mathematical Olympiad
5
The product of pairwise gcd-s is a perfect square
The product of pairwise gcd-s is a perfect square
Source: VI Caucasus Mathematical Olympiad
March 14, 2021
number theory
Problem Statement
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be positive integers such that the product
gcd
(
a
,
b
)
⋅
gcd
(
b
,
c
)
⋅
gcd
(
c
,
a
)
\gcd(a,b) \cdot \gcd(b,c) \cdot \gcd(c,a)
g
cd
(
a
,
b
)
⋅
g
cd
(
b
,
c
)
⋅
g
cd
(
c
,
a
)
is a perfect square. Prove that the product
lcm
(
a
,
b
)
⋅
lcm
(
b
,
c
)
⋅
lcm
(
c
,
a
)
\operatorname{lcm}(a,b) \cdot \operatorname{lcm}(b,c) \cdot \operatorname{lcm}(c,a)
lcm
(
a
,
b
)
⋅
lcm
(
b
,
c
)
⋅
lcm
(
c
,
a
)
is also a perfect square.
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