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Thailand Online MO
2021 Thailand Online MO
P8
P8
Part of
2021 Thailand Online MO
Problems
(1)
Functional equation with gcd
Source: 2021 Thailand Online MO P8 (Mock TMO contest)
4/6/2021
Let
N
\mathbb N
N
be the set of positive integers. Determine all functions
f
:
N
×
N
→
N
f:\mathbb N\times\mathbb N\to\mathbb N
f
:
N
×
N
→
N
that satisfy both of the following conditions:[*]
f
(
gcd
(
a
,
b
)
,
c
)
=
gcd
(
a
,
f
(
c
,
b
)
)
f(\gcd (a,b),c) = \gcd (a,f(c,b))
f
(
g
cd
(
a
,
b
)
,
c
)
=
g
cd
(
a
,
f
(
c
,
b
))
for all
a
,
b
,
c
∈
N
a,b,c \in \mathbb{N}
a
,
b
,
c
∈
N
. [*]
f
(
a
,
a
)
≥
a
f(a,a) \geq a
f
(
a
,
a
)
≥
a
for all
a
∈
N
a \in \mathbb{N}
a
∈
N
.
functional equation
number theory
greatest common divisor