Prove that cube of any positive integer can be written as 3m,3m+1 or 3

1.	Prove that cube of any positive integer can be written as 3m,3m+1 or 3m+2
Answer» Let, \'n\' be a positive integer.By using Euclid\'s Division Lemma,\'n\' can be written in the form of—3q+r, where 0 <= r <3.Therefore, r = {0,1,2}Case 1:r = 0n = 3q=>n³ = 27q³ = 3m (where m = 9q³)Case 2:r = 1n = 3q+1=>n³ = 27q³+27q²+9q+1 = 3m +1(where m = 9q³+9q²+3q)Case 3:r = 2n = 3q+2=>n³ = 27q³+54q²+36q+8 = 3m+2(where m = 9q³+18q²+12q+2)Hence, it is proved that the cube of any positive integer can be written in the form of 3m or 3m+1 or 3m+2. ORWe know that the cube of any positive integer is a positive integer.On applying Euclid\'s division lemma,n³ = 3m+r, where 0 ≤ r < 3Therefore, r = {0,1,2}Therefore, n³ = 3m or 3m+1 or 3m+2 Ex: agar tum 3m le rahe ho vaha pe eg taur par 3q lena aur use square kar \'q\' se aage jo bhi tumhe milega use \'m\' consider karo aur doubt katam

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