# Periodically problematic

Algebra Level 5

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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