Show that the square of any positive integer cannot be of the form 6m+2 or 6m+5 for some integer q.

Explanation

To show that the square of any positive integer cannot be of the form \(6m + 2\) or \(6m + 5\) for some integer \(m\), we will analyze the possible forms of a positive integer when divided by 6.

### Step 1: Represent any integer in the form of \(6m + b\)
Any integer can be expressed in the form \(6m + b\), where \(b\) can take values \(0, 1, 2, 3, 4, 5\).

### Step 2: Calculate the square for each case of \(b\)
Now, we will calculate the square of \(q = 6m + b\) for each possible value of \(b\):

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