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Contests
National and Regional Contests
China Contests
China Team Selection Test
2012 China Team Selection Test
2
find all k
find all k
Source: 2012 China TST Test 3 p5
March 26, 2012
number theory proposed
number theory
Problem Statement
Find all integers
k
≥
3
k\ge 3
k
≥
3
with the following property: There exist integers
m
,
n
m,n
m
,
n
such that
1
<
m
<
k
1<m<k
1
<
m
<
k
,
1
<
n
<
k
1<n<k
1
<
n
<
k
,
gcd
(
m
,
k
)
=
gcd
(
n
,
k
)
=
1
\gcd (m,k)=\gcd (n,k) =1
g
cd
(
m
,
k
)
=
g
cd
(
n
,
k
)
=
1
,
m
+
n
>
k
m+n>k
m
+
n
>
k
and
k
∣
(
m
−
1
)
(
n
−
1
)
k\mid (m-1)(n-1)
k
∣
(
m
−
1
)
(
n
−
1
)
.
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