Let f be a function with
(principal-)period
fractionReduce(Z1,N1).
What is the (principal-)period of the function
{\color{blue}g} with
{\color{blue}g(x) =
f\left(fractionReduce(Z,N) \cdot x \right)}
?
We are looking for the smallest number
{\color{red}P} > 0
with
{\color{blue}g(x + {\color{red}P}) = g(x)}.
It is {\color{blue}g(x + {\color{red}P})}
=
f \left(
fractionReduce(Z,N)
(x +{\color{red}P})\right) =
f \left(
fractionReduce(Z,N) x +
fractionReduce(Z,N) {\color{red}P}
\right).
With given period
\color{orange}{fractionReduce(Z1,N1)}
of f we try to choose {\color{red}P} such that
\color{orange}{
fractionReduce(Z,N) \cdot {\color{red}P}
= fractionReduce(Z1,N1)}
.
Because this guarantees
f \left(
fractionReduce(Z,N) x +
\color{orange}{
fractionReduce(Z,N) \cdot
{\color{red}P}}
\right) =
f \left(fractionReduce(Z,N) x +
\color{orange}{fractionReduce(Z1,N1)}
\right) =
f \left(fractionReduce(Z,N) x \right)
={\color{blue}g(x)}.
To {\color{red}P} get it, solve
the equation \color{orange}{
fractionReduce(Z,N) \cdot {\color{red}P}
= fractionReduce(Z1,N1)}
for {\color{red}P} and find
\displaystyle
{\color{red}P} =
fractionReduce(N*Z1,Z*N1).