Show that the cube of a positive integer of the form `6q+r,q` is an integer and r=0,1,2,3,4,5 is also of the form 6m+r

Explanation

To show that the cube of a positive integer of the form \(6q + r\) (where \(q\) is an integer and \(r = 0, 1, 2, 3, 4, 5\)) is also of the form \(6m + r\) for some integer \(m\), we will analyze each case of \(r\) separately.

### Step-by-Step Solution:

1. **Express the Positive Integer**:
Let \(n\) be a positive integer of the form \(n = 6q + r\), where \(q\) is a non-negative integer and \(r\) can take values \(0, 1, 2, 3, 4, 5\).

2. **Cube the Integer**:
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