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Tuymaada Olympiad
2019 Tuymaada Olympiad
6
Fun easy number theory
Fun easy number theory
Source: Tuymaada 2019 Senior P5 out of 8
July 15, 2019
number theory
Fun
prime numbers
Problem Statement
Let
S
\mathbb{S}
S
is the set of prime numbers that less or equal to 26. Is there any
a
1
,
a
2
,
a
3
,
a
4
,
a
5
,
a
6
∈
N
a_1, a_2, a_3, a_4, a_5, a_6 \in \mathbb{N}
a
1
,
a
2
,
a
3
,
a
4
,
a
5
,
a
6
∈
N
such that
g
c
d
(
a
i
,
a
j
)
∈
S
for
1
≤
i
≠
j
≤
6
gcd(a_i,a_j) \in \mathbb{S} \qquad \text {for } 1\leq i \ne j \leq 6
g
c
d
(
a
i
,
a
j
)
∈
S
for
1
≤
i
=
j
≤
6
and for every element
p
p
p
of
S
\mathbb{S}
S
there exists a pair of
1
≤
k
≠
l
≤
6
1\leq k \ne l \leq 6
1
≤
k
=
l
≤
6
such that
s
=
g
c
d
(
a
k
,
a
l
)
?
s=gcd(a_k,a_l)?
s
=
g
c
d
(
a
k
,
a
l
)?
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