Periodically problematic

Algebra Level 5

For how many positive integers kk, does there exists a non-constant function fkf_k from the reals to the reals, which is periodic with fundamental period kk, and for all real values of xx satisfies the equation fk(x30)+fk(x+600)=0? f_k(x - 30 ) + f_k( x + 600 ) = 0?

Details and assumptions

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

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