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2018-IMOC
N2
lcm and gcd FE
lcm and gcd FE
Source: IMOC 2018 N2
August 17, 2021
number theory
least common multiple
greatest common divisor
fe
functional equation
Problem Statement
Find all functions
f
:
N
→
N
f:\mathbb N\to\mathbb N
f
:
N
→
N
satisfying
lcm
(
f
(
x
)
,
y
)
gcd
(
f
(
x
)
,
f
(
y
)
)
=
f
(
x
)
f
(
f
(
y
)
)
\operatorname{lcm}(f(x),y)\gcd(f(x),f(y))=f(x)f(f(y))
lcm
(
f
(
x
)
,
y
)
g
cd
(
f
(
x
)
,
f
(
y
))
=
f
(
x
)
f
(
f
(
y
))
for all
x
,
y
∈
N
x,y\in\mathbb N
x
,
y
∈
N
.
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