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Taiwan Contests
Taiwan National Olympiad
1997 Taiwan National Olympiad
7
7
Part of
1997 Taiwan National Olympiad
Problems
(1)
Find all positive integers $k$
Source: 6-th Taiwanese Mathematical Olympiad 1997
1/17/2007
Find all positive integers
k
k
k
for which there exists a function
f
:
N
→
Z
f: \mathbb{N}\to\mathbb{Z}
f
:
N
→
Z
satisfying
f
(
1997
)
=
1998
f(1997)=1998
f
(
1997
)
=
1998
and
f
(
a
b
)
=
f
(
a
)
+
f
(
b
)
+
k
f
(
gcd
(
a
,
b
)
)
∀
a
,
b
f(ab)=f(a)+f(b)+kf(\gcd{(a,b)})\forall a,b
f
(
ab
)
=
f
(
a
)
+
f
(
b
)
+
k
f
(
g
cd
(
a
,
b
)
)
∀
a
,
b
.
function
number theory proposed
number theory