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Baltic Way
2001 Baltic Way
11
11
Part of
2001 Baltic Way
Problems
(1)
Function acting on a,b, and gcd(a,b)
Source: Baltic Way 2001
11/17/2010
The real-valued function
f
f
f
is defined for all positive integers. For any integers
a
>
1
,
b
>
1
a>1, b>1
a
>
1
,
b
>
1
with
d
=
gcd
(
a
,
b
)
d=\gcd (a, b)
d
=
g
cd
(
a
,
b
)
, we have
f
(
a
b
)
=
f
(
d
)
(
f
(
a
d
)
+
f
(
b
d
)
)
f(ab)=f(d)\left(f\left(\frac{a}{d}\right)+f\left(\frac{b}{d}\right)\right)
f
(
ab
)
=
f
(
d
)
(
f
(
d
a
)
+
f
(
d
b
)
)
Determine all possible values of
f
(
2001
)
f(2001)
f
(
2001
)
.
function
algebra proposed
algebra