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Show that cube of any positive integer i...

Show that cube of any positive integer is of the form 4m, 4m+1 or 4m+3, for some integer m.

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To show that the cube of any positive integer is of the form \(4m\), \(4m + 1\), or \(4m + 3\) for some integer \(m\), we can follow these steps: ### Step 1: Define the Positive Integer Let \(a\) be any positive integer. According to the division algorithm, we can express \(a\) in the form: \[ a = 4q + r \] where \(q\) is some integer, and \(r\) is the remainder when \(a\) is divided by 4. The possible values for \(r\) are \(0\), \(1\), \(2\), or \(3\). ...
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