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National and Regional Contests
PEN Problems
PEN E Problems
17
17
Part of
PEN E Problems
Problems
(1)
E 17
Source:
5/25/2007
Let
a
a
a
,
b
b
b
, and
n
n
n
be positive integers with
gcd
(
a
,
b
)
=
1
\gcd (a, b)=1
g
cd
(
a
,
b
)
=
1
. Without using Dirichlet's theorem, show that there are infinitely many
k
∈
N
k \in \mathbb{N}
k
∈
N
such that
gcd
(
a
k
+
b
,
n
)
=
1
\gcd(ak+b, n)=1
g
cd
(
ak
+
b
,
n
)
=
1
.
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