Question

Use Euclid division lemma show that cube of any positive integer is of the form 9m,9m+1,9m+2

Answer 1

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, x3 is either of the form 9 m or 9 m + 1 or, 9 m + 8.

Answer 2

Let a be any positive integer and b = 3 a = 3q + r, where q ≥ 0 and 0 ≤ r < 3 a = 3q or 3q + 1 or 3q + 2 Therefore, every number can be represented as these three forms. There are three cases.

Case 1: When a = 3q, a3 = (3q)3 = 27q3 = 9(3q3)= 9m Where m is an integer such that m = 3q3

Case 2: When a = 3q + 1, a3 = (3q +1)3 a3 = 27q3 + 27q2 + 9q + 1 a3 = 9(3q3 + 3q2 + q) + 1 a3 = 9m + 1 Where m is an integer such that m = (3q3 + 3q2 + q)

Case 3: When a = 3q + 2, a3 = (3q +2)3 a3 = 27q3 + 54q2 + 36q + 8 a3 = 9(3q3 + 6q2 + 4q) + 8 a3 = 9m + 8 Where m is an integer such that m = (3q3 + 6q2 + 4q) Therefore, the cube of any positive integer is of the form 9m, 9m + 1, or 9m +8 use-euclids-division-lemma-to-show-that-the-cube-of-any-positive-integer-is-of-the-form-9m-9m.

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