Use Euclid’s division lemma to show that the cube of any positive integer is of the form 9m, 9m + 1 or 9m + 8.

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Answer,
Let a be any positive integer and b = 3, using Euclid’s Division Lemma, a = 3q + r, where q ≥ 0 and 0 ≤ r < 3. Therefore, a = 3q or 3q + 1 or 3q + 2.
Therefore, every number can be represented as these three forms.
There are three cases.
Case 1: When a = 3q,
a³ = (3q)³ = 27q³ = 9(3q³)= 9m
Where m is an integer such that m = 3q³
Case 2: When a = 3q + 1,
a³ = (3q +1)³
a³ = 27q³ + 27q² + 9q + 1
a³ = 9(3q³ + 3q² + q) + 1 = 9m + 1
Where m is an integer such that m = (3q³ + 3q² + q)
Case 3: When a = 3q + 2,
a³ = (3q +2)³
a³ = 27q³ + 54q² + 36q + 8
a³ = 9(3q³ + 6q² + 4q) + 8
a³ = 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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