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Cuba MO
2005 Cuba MO
7
gcd(x, y) = 6, gcd(y, z) = 10, gcd(z, x) = 8, lcm(x, y, z) =2400
gcd(x, y) = 6, gcd(y, z) = 10, gcd(z, x) = 8, lcm(x, y, z) =2400
Source: 2005 Cuba MO 2.7
September 15, 2024
number theory
LCM
GCD
GCD and LCM
Problem Statement
Determine all triples of positive integers
(
x
,
y
,
z
)
(x, y, z)
(
x
,
y
,
z
)
that satisfy
x
<
y
<
z
,
g
c
d
(
x
,
y
)
=
6
,
g
c
d
(
y
,
z
)
=
10
,
g
c
d
(
z
,
x
)
=
8
a
n
d
l
c
m
(
x
,
y
,
z
)
=
2400.
x < y < z, \ \ gcd(x, y) = 6, \ \ gcd(y, z) = 10, \ \ gcd(z, x) = 8 \ \ and \ \ lcm(x, y, z) = 2400.
x
<
y
<
z
,
g
c
d
(
x
,
y
)
=
6
,
g
c
d
(
y
,
z
)
=
10
,
g
c
d
(
z
,
x
)
=
8
an
d
l
c
m
(
x
,
y
,
z
)
=
2400.
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