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Korea Contests
Korea National Olympiad
2013 Korea National Olympiad
5
5
Part of
2013 Korea National Olympiad
Problems
(1)
Functional Equation with gcd and lcm
Source: Korea National 2013 #5
11/10/2013
Find all functions
f
:
N
→
N
f : \mathbb{N} \rightarrow \mathbb{N}
f
:
N
→
N
satisfying
f
(
m
n
)
=
lcm
(
m
,
n
)
⋅
gcd
(
f
(
m
)
,
f
(
n
)
)
f(mn) = \operatorname{lcm} (m,n) \cdot \gcd( f(m), f(n) )
f
(
mn
)
=
lcm
(
m
,
n
)
⋅
g
cd
(
f
(
m
)
,
f
(
n
))
for all positive integer
m
,
n
m,n
m
,
n
.
number theory
greatest common divisor
least common multiple
function
number theory proposed