Question

Use Euclid Lemma to show that the cube of any positive integer is of the form 9 m, 9 M + 1 or 9 M + 8

Answer 1

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,

Where m is an integer such that m =
Case 2: When a = 3q + 1,
a ³ = (3q +1) ³
a ³ = 27q ³ + 27q ² + 9q + 1
a ³ = 9(3q ³ + 3q ² + q) + 1
a ³ = 9m + 1
Where m is an integer such that m = (3q ³ + 3q ² + 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 = (3q ³ + 6q ² + 4q)
Therefore, the cube of any positive integer is of the form 9m, 9m + 1, or 9m + 8.