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Caucasus Mathematical Olympiad
2023 Caucasus Mathematical Olympiad
6
\gcd(a, b) + lcm(a, b) = \gcd(a, c) + lcm(a, c)
\gcd(a, b) + lcm(a, b) = \gcd(a, c) + lcm(a, c)
Source: Caucasus MO 2023
July 16, 2023
number theory
Problem Statement
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be positive integers such that
gcd
(
a
,
b
)
+
lcm
(
a
,
b
)
=
gcd
(
a
,
c
)
+
lcm
(
a
,
c
)
.
\gcd(a, b) + \text{lcm}(a, b) = \gcd(a, c) + \text{lcm}(a, c).
g
cd
(
a
,
b
)
+
lcm
(
a
,
b
)
=
g
cd
(
a
,
c
)
+
lcm
(
a
,
c
)
.
Does it follow from this that
b
=
c
b = c
b
=
c
?
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