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Czech Republic Contests
Czech and Slovak Olympiad III A
2024 Czech and Slovak Olympiad III A
1
1
Part of
2024 Czech and Slovak Olympiad III A
Problems
(1)
gcd(a,b) x lcm(b,c), gcd(b,c)x lcm(c,a), ,gcd(c,a) x lcm(a,b)
Source: 2024 Czech and Slovak Olympiad III A p1
5/18/2024
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be positive integers such that one of the values
g
c
d
(
a
,
b
)
⋅
l
c
m
(
b
,
c
)
,
g
c
d
(
b
,
c
)
⋅
l
c
m
(
c
,
a
)
,
g
c
d
(
c
,
a
)
−
⋅
l
c
m
(
a
,
b
)
gcd(a,b) \cdot lcm(b,c), \,\,\,\, gcd(b,c)\cdot lcm(c,a), \,\,\,\, gcd(c,a)-\cdot lcm(a,b)
g
c
d
(
a
,
b
)
⋅
l
c
m
(
b
,
c
)
,
g
c
d
(
b
,
c
)
⋅
l
c
m
(
c
,
a
)
,
g
c
d
(
c
,
a
)
−
⋅
l
c
m
(
a
,
b
)
is equal to the product of the remaining two. Prove that one of the numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
is a multiple of another of them.
number theory
GCD
LCM