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Baltic Way
2019 Baltic Way
16
16
Part of
2019 Baltic Way
Problems
(1)
Number of solutions is always a perfect square
Source: 2019 Baltic Way P16
11/18/2019
For a positive integer
N
N
N
, let
f
(
N
)
f(N)
f
(
N
)
be the number of ordered pairs of positive integers
(
a
,
b
)
(a,b)
(
a
,
b
)
such that the number
a
b
a
+
b
\frac{ab}{a+b}
a
+
b
ab
is a divisor of
N
N
N
. Prove that
f
(
N
)
f(N)
f
(
N
)
is always a perfect square.
number theory