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Use Euclid's division lemma to show tha...

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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To show that the cube of any positive integer is of the form \(9m\), \(9m+1\), or \(9m+8\) using Euclid's Division Lemma, we will follow these steps: ### Step 1: Understand Euclid's Division Lemma According to Euclid's Division Lemma, for any integer \(A\) and a positive integer \(B\), there exist unique integers \(Q\) (the quotient) and \(R\) (the remainder) such that: \[ A = BQ + R \] where \(0 \leq R < B\). In this case, we will let \(B = 3\). ...
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