MathDB
Problems
Contests
National and Regional Contests
Turkey Contests
National Olympiad First Round
1999 National Olympiad First Round
18
18
Part of
1999 National Olympiad First Round
Problems
(1)
Number Theory
Source: 0
4/21/2009
Let
t
k
(
n
)
t_{k} \left(n\right)
t
k
(
n
)
show the sum of
k
t
h
k^{th}
k
t
h
power of digits of positive number
n
n
n
. For which
k
k
k
, the condition that
t
k
(
n
)
t_{k} \left(n\right)
t
k
(
n
)
is a multiple of 3 does not imply the condition that
n
n
n
is a multiple of 3?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
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x
−
b
o
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>
(
A
)
<
/
s
p
a
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>
3
<
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c
l
a
s
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=
′
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a
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x
−
b
o
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>
(
B
)
<
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>
6
<
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c
l
a
s
s
=
′
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x
−
b
o
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>
(
C
)
<
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>
9
<
s
p
a
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c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
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a
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>
15
<
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c
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a
s
s
=
′
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a
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b
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>
(
E
)
<
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>
None
<span class='latex-bold'>(A)</span>\ 3 \qquad<span class='latex-bold'>(B)</span>\ 6 \qquad<span class='latex-bold'>(C)</span>\ 9 \qquad<span class='latex-bold'>(D)</span>\ 15 \qquad<span class='latex-bold'>(E)</span>\ \text{None}
<
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p
an
c
l
a
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=
′
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a
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x
−
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>
(
A
)
<
/
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p
an
>
3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
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p
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>
6
<
s
p
an
c
l
a
ss
=
′
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a
t
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x
−
b
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′
>
(
C
)
<
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>
9
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
15
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
None