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2024 Japan MO Finals
2
Function about LCM
Function about LCM
Source: 2024 Japan MO P2
February 11, 2024
number theory
functional equation
least common multiple
induction
Problem Statement
Find all functions
f
:
Z
>
0
→
Z
>
0
f:\mathbb{Z}_{>0}\rightarrow\mathbb{Z}_{>0}
f
:
Z
>
0
→
Z
>
0
such that
lcm
(
m
,
f
(
m
+
f
(
n
)
)
)
=
lcm
(
f
(
m
)
,
f
(
m
)
+
n
)
\text{lcm}(m, f(m+f(n)))=\text{lcm}(f(m), f(m)+n)
lcm
(
m
,
f
(
m
+
f
(
n
)))
=
lcm
(
f
(
m
)
,
f
(
m
)
+
n
)
for any positive integers
m
m
m
and
n
n
n
.
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