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the cube of any positive integer is either of the form 4m or 4m+1 or 4m+ 3
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Answers
Solution:-
Let 'a' be any positive Integer and 'b = 4'.
∴ By Euclid Division Lemma,
The possible values of r will be 1 , 2 and 3.
=) 'a' can takes the value of :- 4q , (4q + 1) , (4q + 2) and (4q + 3).
Case I,
a = 4q
Cubing on both the sides.
=) a³ = ( 4q)³
=) a³ = 64q³
=) a³ = 4 ( 16q³)
Taking [ a³ = x ] and [ 16q³ = m ]
=) x = 4m
Case II,
a = 4q + 1
Cubing on both the sides
=) a³ = ( 4q + 1)³
=) a³ = ( 4q)³ + 1 + 12q ( 4q + 1)
=) a³ = 64q³ + 1 + 48q² + 12q
=) a³ = 64q³ + 48q² + 12q + 1
=) a³ = 4 ( 16q³ + 12q² + 3q ) + 1
Taking [ a³ = x ] and [ 16q³ + 12q² + 3q = m ]
=) x = 4m + 1
Case III,
a = ( 4q + 2)
Cubing on both the sides
=) a³ = ( 4q + 2)³
=) a³ = ( 4q)³ + 2³ + 24q ( 4q + 2)
=) a³ = 64q³ + 8 + 96q² + 48q
=) a³ = 64q³ + 96q² + 48q + 8
=) a³ = 4 ( 16q³ + 24q² + 12q + 2 )
Taking [ a³ = x ] and [ 16q³ + 24q² + 12q + 2 = m ]
=) x = 4m
Case IV,
a = 4q + 3
Cubing on both the sides.
=) a³ = ( 4q + 3)³
=) a³ = 64q³ + 27 + 36q ( 4q + 3)
=) a³ = 64q³ + 27 + 144q² + 108q
=) a³ = 64q³ + 144q² + 108q + 24 + 3
=) a³ = 4 ( 16q³ + 36q² + 27q + 6 ) + 3
Taking [ a³ = x ] and [ 16q³ + 36q² + 27q + 6 = m ]
=) x = 4m + 3.
∴ From all the cases , The cube of any positive integer is either of the form 4m or 4m+1 or 4m+ 3.