Using Euclid division leema. Show that the qube of any odd integer is

1.	Using Euclid division leema. Show that the qube of any odd integer is 9q+1 or 9q+8
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,\xa0Where m is an integer such that m = Case 2: When a = 3q + 1,a\xa03\xa0= (3q +1)\xa03\xa0a\xa03\xa0= 27q\xa03\xa0+ 27q\xa02\xa0+ 9q + 1\xa0a\xa03\xa0= 9(3q\xa03\xa0+ 3q\xa02\xa0+ q) + 1a\xa03\xa0= 9m + 1\xa0Where m is an integer such that m = (3q\xa03\xa0+ 3q\xa02\xa0+ q)\xa0Case 3: When a = 3q + 2,a\xa03\xa0= (3q +2)\xa03\xa0a\xa03\xa0= 27q\xa03\xa0+ 54q\xa02\xa0+ 36q + 8\xa0a\xa03\xa0= 9(3q\xa03\xa0+ 6q\xa02\xa0+ 4q) + 8a\xa03\xa0= 9m + 8Where m is an integer such that m = (3q\xa03\xa0+ 6q\xa02\xa0+ 4q)\xa0Therefore, the cube of any positive integer is of the form 9m, 9m + 1, or 9m + 8.

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