# Use euclids division lemma to show that the cube of any positive integer is of the form of 9m,9m+1 or 9m+8

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x = 3q ,or 3q + 1, or 3q +2

so, x³ = (3q)³

= 27q³

= 9(3q³)

=9m where m = 3q³

similarly, x³ = (3q +1)³

= 27q³ + 1³ + 3(3q²)(1) + 3(3q)(1)²

= 27q³ + 1 + 27q² + 9q

= 27q³ + 27q² + 9q + 1

= 9[q(3q² + 3q +1)] + 1

= 9m + 1 where m= q(3q² +3q + 1)

again, x³ = (3q +2)³

= 27q³ + 2³ + 3(3q)²(2) + 3(3q)(2)²

= 27q³ + 8 + 54q² + 36q

= 27q³ + 54q² + 36q + 8

= 9[q(3q² + 6q + 4)] + 8

= 9m + 8 where m= q(3q² + 6q + 4)