InterviewSolution
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InterviewSolution
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5. Use Euclid's division lemma to show that the cube of any positive integer is oft9m, 9m + 1 or 9m +8. |
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Answer» Let a be any positive integer and b = 3a = 3q + r, where q ≥ 0 and 0 ≤ r < 3Therefore, every number can be represented as these three forms. There are three cases. Case 1: When a = 3q,Where m is an integer such that m = (3q)3 = 27q39(3q3) = 9m Case 2: When a = 3q + 1,a3= (3q +1)3a3= 27q3+ 27q2+ 9q + 1a3= 9(3q3+ 3q2+ q) + 1a3= 9m + 1Where m is an integer such that m = (3q3+ 3q2+ q) Case 3: When a = 3q + 2,a3= (3q +2)3a3= 27q3+ 54q2+ 36q + 8a3= 9(3q3+ 6q2+ 4q) + 8a3= 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. |
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