The roots of the quadratic equation \(x^2-35x+300=0\) are \(\alpha\) and \(\beta\). The quadratic equation which has only one real root \(\alpha^3+\beta^3\) can be expressed as \(x^2-mx+n=0\), where \(m\) and \(n\) are positive integers.

Find the sum of digits of \(n\) without using a calculator.

**Bonus**: Solve this problem without finding the values of \(\alpha\) and \(\beta\).

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