Use Euclid's division lemma to show that the cube of a positive intege

1.	Use Euclid's division lemma to show that the cube of a positive integers is of that form 9m,9m+1or9m+8
Answer» Let x be any positive integer. Then, it is of the form 3q or, 3q + 1 or, 3q + 2. So, we have the following cases : Case I : When x = 3q. then, x³ = (3q)³ = 27q³ = 9 (3q³) = 9m, where m = 3q³. Case II : When x = 3q + 1 then, x³ = (3q + 1)³ = 27q³ + 27q² + 9q + 1 = 9 q (3q² + 3q + 1) + 1 = 9m + 1, where m = q (3q² + 3q + 1) Case III. When x = 3q + 2 then, x³ = (3q + 2)³ = 27 q³ + 54q² + 36q + 8 = 9q (3q² + 6q + 4) + 8 = 9 m + 8, where m = q (3q² + 6q + 4) Hence, x³is either of the form 9 m or 9 m + 1 or, 9 m + 8.

Use Euclid's division lemma to show that the cube of a positive integers is of that form 9m,9m+1or9m+8

Answer»

Let x be any positive integer. Then, it is of the form 3q or, 3q + 1 or, 3q + 2.

So, we have the following cases :

Case I : When x = 3q.

then, x³ = (3q)³ = 27q³ = 9 (3q³) = 9m, where m = 3q³.

Case II : When x = 3q + 1

then, x³ = (3q + 1)³

= 27q³ + 27q² + 9q + 1

= 9 q (3q² + 3q + 1) + 1

= 9m + 1, where m = q (3q² + 3q + 1)

Case III. When x = 3q + 2

then, x³ = (3q + 2)³

= 27 q³ + 54q² + 36q + 8

= 9q (3q² + 6q + 4) + 8

= 9 m + 8, where m = q (3q² + 6q + 4)

Hence, x³is either of the form 9 m or 9 m + 1 or, 9 m + 8.

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