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Romanian Masters of Mathematics Collection
2018 Romanian Master of Mathematics
4
4
Part of
2018 Romanian Master of Mathematics
Problems
(1)
Number theory
Source: RMM 2018 D2 P4
2/25/2018
Let
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
be positive integers such that
a
d
≠
b
c
ad \neq bc
a
d
=
b
c
and
g
c
d
(
a
,
b
,
c
,
d
)
=
1
gcd(a,b,c,d)=1
g
c
d
(
a
,
b
,
c
,
d
)
=
1
. Let
S
S
S
be the set of values attained by
gcd
(
a
n
+
b
,
c
n
+
d
)
\gcd(an+b,cn+d)
g
cd
(
an
+
b
,
c
n
+
d
)
as
n
n
n
runs through the positive integers. Show that
S
S
S
is the set of all positive divisors of some positive integer.
number theory