A pair of positive integers \((m, n)\) is called **good** if \(m\) divides \(n^2+1\) and \(n\) divides \(m^2+1.\)

A positive integer \(n\) is called **admissible** if there exists an integer \(m\) for which the pair \((m, n)\) is good.

Find the sum of all admissible positive integers \(\leq 100.\)

**Details and assumptions**

- As an explicit example, since \(5\) divides \(13^2+1\) and \(13\) divides \(5^2+1,\) the pair \((5, 13)\) is good. This also implies \(5\) are \(13\) are admissible integers.

- This problem is an extension of an old BMO problem.

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