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KVPY SX SB Exam 2015 Question Paper
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© examsnet.com
Question : 3
Total: 120
The largest perfect square that divides
2014
3
−
2013
3
+
2012
3
−
2011
3
+
.
.
.
.
+
2
3
−
1
3
is -
1
2
2
2
1007
2
2014
2
Validate
Solution:
2
{
(
2014
)
3
+
(
2012
)
2
+
…
+
2
3
–
(
2014
)
3
+
(
2013
)
3
+
…
+
1
3
}
=
2
×
8
{
(
1007
)
2
+
(
1006
)
2
+
…
+
1
3
}
−
{
(
2014
)
3
+
(
2013
)
2
+
…
+
1
3
}
=
2
×
8
×
(
(
1007
)
(
1008
)
2
)
2
−
(
(
2014
)
(
2015
)
2
)
2
=
2
×
8
×
(
1007
)
2
(
1008
)
2
4
−
(
2014
)
2
(
2015
)
2
4
=
(
1007
)
2
(
2016
)
2
−
(
1007
)
2
(
2015
)
2
=
(
1007
)
2
{
2016
–
2015
}
{
2016
+
2015
}
=
(
1007
)
2
(
4031
)
= divisible by
(
1007
)
2
© examsnet.com
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