Write whether the square of any positive integer can be of the form 3m+2, where m is a natural number. Justify answer.

Explanation

To determine whether the square of any positive integer can be of the form \(3m + 2\) (where \(m\) is a natural number), we can analyze the possible values of the square of a positive integer modulo 3.

### Step-by-step Solution:

1. **Understanding the Problem**: We need to check if \(n^2\) (where \(n\) is a positive integer) can equal \(3m + 2\) for some natural number \(m\). This means we need to see if \(n^2 \equiv 2 \mod 3\).

2. **Possible Remainders**: When a positive integer \(n\) is divided by 3, the possible remainders (or residues) are 0, 1, or 2. We will check the square of \(n\) for each of these cases.

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