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Byju's Answer
Standard IX
Mathematics
Natural Numbers
Prove that th...
Question
Prove that the fraction
m
(
n
+
1
)
+
1
m
(
n
+
1
)
−
n
is irreducible for every positive integers
m
and
n
.
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Solution
Let us consider three cases as follows:
Case 1: When
m
is even and
n
is even, then
m
=
2
p
and
n
=
2
q
.
Therefore,
m
(
n
+
1
)
+
1
m
(
n
+
1
)
−
n
=
2
p
(
2
q
+
1
)
+
1
2
p
(
2
q
+
1
)
−
2
q
=
4
p
q
+
2
p
+
1
4
p
q
+
2
p
−
2
q
which is not reducible.
Case 2: When
m
is even and
n
is odd, then
m
=
2
p
and
n
=
2
q
+
1
.
Therefore,
m
(
n
+
1
)
+
1
m
(
n
+
1
)
−
n
=
2
p
(
2
q
+
1
+
1
)
+
1
2
p
(
2
q
+
1
+
1
)
−
(
2
q
+
1
)
=
2
p
(
2
q
+
2
)
+
1
2
p
(
2
q
+
2
)
−
2
q
−
1
=
4
p
q
+
4
p
+
1
4
p
q
+
4
p
−
2
q
−
1
which is not reducible.
Case 3: When
m
is odd and
n
is odd, then
m
=
2
p
+
1
and
n
=
2
q
+
1
.
Therefore,
m
(
n
+
1
)
+
1
m
(
n
+
1
)
−
n
=
(
2
p
+
1
)
(
2
q
+
1
+
1
)
+
1
(
2
p
+
1
)
(
2
q
+
1
+
1
)
−
(
2
q
+
1
)
=
(
2
p
+
1
)
(
2
q
+
2
)
+
1
(
2
p
+
1
)
(
2
q
+
2
)
−
2
q
−
1
=
4
p
q
+
4
p
+
2
q
+
2
+
1
4
p
q
+
4
p
+
2
q
+
2
−
2
q
−
1
=
4
p
q
+
4
p
+
2
q
+
3
4
p
q
+
4
q
+
1
which is not reducible.
Hence the fraction
m
(
n
+
1
)
+
1
m
(
n
+
1
)
−
n
is irreducible for every positive integer
m
and
n
.
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