Use Euclid division Lemma to show that cube of any positive integer is

1.	Use Euclid division Lemma to show that cube of any positive integer is form of 99 + 19 + 8
Answer» Q:\xa0Use Euclid\'s division lemma to show that the cube of any positive integer is of the form\xa09m,9m+1\xa0or\xa09m+8Let\xa0x\xa0be any positive integer. Then, it is of the form 3q\xa0or, 3q\xa0+ 1 or, 3q\xa0+ 2.So, we have the following cases :\xa0Case I :\xa0When\xa0x\xa0= 3q.then,\xa0x3\xa0= (3q)3\xa0= 27q3\xa0= 9 (3q3) = 9m, where\xa0m\xa0= 3q3.\xa0Case II :\xa0When\xa0x\xa0= 3q\xa0+ 1then,\xa0x3\xa0= (3q\xa0+ 1)3= 27q3\xa0+ 27q2\xa0+ 9q\xa0+ 1= 9\xa0q\xa0(3q2\xa0+ 3q\xa0+ 1) + 1= 9m\xa0+ 1, where\xa0m\xa0=\xa0q\xa0(3q2\xa0+ 3q\xa0+ 1)\xa0Case III.\xa0When\xa0x\xa0= 3q\xa0+ 2then,\xa0x3\xa0= (3q\xa0+ 2)3= 27\xa0q3\xa0+ 54q2\xa0+ 36q\xa0+ 8= 9q\xa0(3q2\xa0+ 6q\xa0+ 4) + 8= 9\xa0m\xa0+ 8, where\xa0m\xa0=\xa0q\xa0(3q2\xa0+ 6q\xa0+ 4)Hence,\xa0x3\xa0is either of the form 9\xa0m\xa0or 9\xa0m\xa0+ 1 or, 9\xa0m\xa0+ 8.

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