For how many positive integers \(k\), does there exists a non-constant function \(f_k\) from the reals to the reals, which is periodic with fundamental period \(k\), and for all real values of \(x\) satisfies the equation \[ f_k(x - 30 ) + f_k( x + 600 ) = 0? \]

**Details and assumptions**

The **fundamental period** of a non-constant function \(f\) on the reals is the smallest non-negative value \(\alpha\) such that \( f(x) = f( x + \alpha) \) for all real \(x\).

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