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Source:
January 1, 2022
number theory
Problem Statement
Given two positive integers
a
≠
b
a \ne b
a
=
b
, let
f
(
a
,
b
)
f(a, b)
f
(
a
,
b
)
be the smallest integer that divides exactly one of
a
,
b
a, b
a
,
b
, but not both. Determine the number of pairs of positive integers
(
x
,
y
)
(x, y)
(
x
,
y
)
, where
x
≠
y
x \ne y
x
=
y
,
1
≤
x
,
y
,
≤
100
1\le x, y, \le 100
1
≤
x
,
y
,
≤
100
and
gcd
(
f
(
x
,
y
)
,
gcd
(
x
,
y
)
)
=
2
\gcd(f(x, y), \gcd(x, y)) = 2
g
cd
(
f
(
x
,
y
)
,
g
cd
(
x
,
y
))
=
2
.
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