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Problems
Contests
National and Regional Contests
Turkey Contests
National Olympiad First Round
2007 National Olympiad First Round
30
30
Part of
2007 National Olympiad First Round
Problems
(1)
Turkey NMO 2007 1st Round - P30 (Number Theory)
Source:
10/5/2012
Let
(
a
n
)
n
=
1
∞
(a_n)_{n=1}^{\infty}
(
a
n
)
n
=
1
∞
be an integer sequence such that
a
n
+
48
≡
a
n
(
m
o
d
35
)
a_{n+48} \equiv a_n \pmod {35}
a
n
+
48
≡
a
n
(
mod
35
)
for every
n
≥
1
n \geq 1
n
≥
1
. Let
i
i
i
and
j
j
j
be the least numbers satisfying the conditions
a
n
+
i
≡
a
n
(
m
o
d
5
)
a_{n+i} \equiv a_n \pmod {5}
a
n
+
i
≡
a
n
(
mod
5
)
and
a
n
+
j
≡
a
n
(
m
o
d
7
)
a_{n+j} \equiv a_n \pmod {7}
a
n
+
j
≡
a
n
(
mod
7
)
for every
n
≥
1
n\geq 1
n
≥
1
. Which one below cannot be an
(
i
,
j
)
(i,j)
(
i
,
j
)
pair?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
(
16
,
4
)
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
(
3
,
16
)
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
(
8
,
6
)
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
(
1
,
48
)
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
(
16
,
18
)
<span class='latex-bold'>(A)</span>\ (16,4) \qquad<span class='latex-bold'>(B)</span>\ (3,16) \qquad<span class='latex-bold'>(C)</span>\ (8,6) \qquad<span class='latex-bold'>(D)</span>\ (1,48) \qquad<span class='latex-bold'>(E)</span>\ (16,18)
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
(
16
,
4
)
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
(
3
,
16
)
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
(
8
,
6
)
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
(
1
,
48
)
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
(
16
,
18
)
modular arithmetic