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National and Regional Contests
Russia Contests
All-Russian Olympiad
1995 All-Russian Olympiad
5
GCD
GCD
Source:
June 30, 2012
number theory
greatest common divisor
Problem Statement
The sequence
a
1
,
a
2
,
.
.
.
a_1, a_2, ...
a
1
,
a
2
,
...
of natural numbers satisfies
G
C
D
(
a
i
,
a
j
)
=
G
C
D
(
i
,
j
)
GCD(a_i, a_j)=GCD(i, j)
GC
D
(
a
i
,
a
j
)
=
GC
D
(
i
,
j
)
for all
i
≠
j
i \neq j
i
=
j
. Prove that
a
i
=
i
a_i=i
a
i
=
i
for all
i
i
i
.
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