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Turkey Team Selection Test
2021 Turkey Team Selection Test
9
Number of elements in a special set
Number of elements in a special set
Source: Turkey 2021 IMO TST Problem 9
May 23, 2021
Turkey
number theory
GCD
Problem Statement
For which positive integer couples
(
k
,
n
)
(k,n)
(
k
,
n
)
, the equality
∣
{
a
∈
Z
+
:
1
≤
a
≤
(
n
k
)
!
,
g
c
d
(
(
a
k
)
,
n
)
=
1
}
∣
=
(
n
k
)
!
6
\Bigg|\Bigg\{{a \in \mathbb{Z}^+: 1\leq a\leq(nk)!, gcd \left(\binom{a}{k},n\right)=1}\Bigg\}\Bigg|=\frac{(nk)!}{6}
{
a
∈
Z
+
:
1
≤
a
≤
(
nk
)!
,
g
c
d
(
(
k
a
)
,
n
)
=
1
}
=
6
(
nk
)!
holds?
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