Use Euclid’s division lemma to show that the cube. of any positive integer is of the form 9m,9m+1 or 9m+8 .
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1. ✎Δnsɯer࿐

   Let us consider a and b where a be any positive number and b is equal to 3.

   According to Euclid’s Division Lemma

   a = bq + r

   So r is an integer greater than or equal to 0 and less than 3.

   Hence r can be either 0, 1 or 2.

   Case 1: When r = 0, the equation becomes

   a = 3q

   Cubing both the sides

   a3 = (3q)3

   a3 = 27 q3

   a3 = 9 (3q3)

   a3 = 9m

   where m = 3q3

   Case 2: When r = 1, the equation becomes

   a = 3q + 1

   Cubing both the sides

   a3 = (3q + 1)3

   a3 = (3q)3 + 13 + 3 × 3q × 1(3q + 1)

   a3 = 27q3 + 1 + 9q × (3q + 1)

   a3 = 27q3 + 1 + 27q2 + 9q

   a3 = 27q3 + 27q2 + 9q + 1

   a3 = 9 ( 3q3 + 3q2 + q) + 1

   a3 = 9m + 1

   Where m = ( 3q3 + 3q2 + q)

   Case 3: When r = 2, the equation becomes

   a = 3q + 2

   Cubing both the sides

   a3 = (3q + 2)3

   a3 = (3q)3 + 23 + 3 × 3q × 2 (3q + 1)

   a3 = 27q3 + 8 + 54q2 + 36q

   a3 = 27q3 + 54q2 + 36q + 8

   a3 = 9 (3q3 + 6q2 + 4q) + 8

   a3 = 9m + 8

   Where m = (3q3 + 6q2 + 4q)therefore a can be any of the form 9m or 9m + 1 or, 9m + 8.

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2. Answer:

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