Many square brackets

Algebra Level 3

t(x)=x+2x+3x++ϕxϕ(ϕ+1)2x t(x) = \lfloor x\rfloor+\lfloor 2x\rfloor+\lfloor 3x\rfloor+\cdots + \lfloor \phi x\rfloor - \frac{\phi(\phi + 1)}2 x

Find the fundamental period of the function t(x)t(x) given that ϕ \phi is a positive integer.

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