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Periodic function from N to R

Source: 2021 Korea Winter Program Test2 Day2 #8

February 7, 2021
function

Problem Statement

For function f:Z+→Rf:\mathbb Z^+ \to \mathbb R and coprime positive integers p,qp,q ; define fp,fqf_p,f_q as fp(x)=f(px)−f(x),fq(x)=f(qx)−f(x)  (x∈Z+)f_p(x)=f(px)-f(x), f_q(x)=f(qx)-f(x) \space \space (x\in\mathbb Z^+) ff satisfies following conditions.
(i)(i) for all rr that isn't multiple of pqpq, f(r)=0f(r)=0
(ii)(ii) ∃m∈Z+\exists m\in \mathbb Z^+ s.t.s.t. ∀x∈Z+,fp(x+m)=fp(x)\forall x\in \mathbb Z^+, f_p(x+m)=f_p(x) and fq(x+m)=fq(x)f_q(x+m)=f_q(x)
Prove that if x≡yx\equiv y (modm)(mod m), then f(x)=f(y)f(x)=f(y) (x,y∈Z+x, y\in \mathbb Z^+).
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