Use Euclid division lemna to show that the cube of a y positive integer is of the form 9 ,9m+1 or 9m+8
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Let a be any positive integer and b = 3
a = 3q + r, where q ≥ 0 and 0 ≤ r < 3
Therefore, every number can be represented as these three forms. There are three cases.
Case 1: When a = 3q,
a³= 27q³
Where m is an integer such that m = 3q³
Case 2: When a = 3q + 1,
a ³ = (3q +1) ³
a ³ = 27q ³+ 27q ² + 9q + 1
a ³ = 9(3q³ + 3q ² + q) + 1
a ³ = 9m + 1
Where m is an integer such that m = (3q³+ 3q² + q)
Case 3: When a = 3q + 2,
a 3 = (3q +2) ³
a 3 = 27q ³ + 54q ²+ 36q + 8
a 3 = 9(3q ³ + 6q ² + 4q) + 8
a 3 = 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.
Hope it helped u.
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