Expressing Integers As A Difference Of Squares

Let \(f(n)\) be defined as the number of ways a positive integer \(n\) can be expressed as \(a^2-b^2\), where \(a\) and \(b\) are positive integers.

Let \(g(n)\) be defined as the smallest possible positive integer \(y\) such that \(f(y)=n\).

If \(M\) is the least common multiple of \(g(1)\), \(g(2)\), \(g(3)\), \(g(4)\), and \(g(5)\), then what is \(f(M)\)?

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