Let f be a function with
(principal-)period
fractionReduce(Z1,N1).
Determine {\color{red}b} such that
{\color{blue}g} with
{\color{blue}g(x) =
f\left({\color{red}b} \cdot x \right)}
has the (prinicpal-)period
fractionReduce(Z,N).
With the desired period
fractionReduce(Z,N)
holds
{\color{blue}
g\left(x + fractionReduce(Z,N)
\right) =
g(x)}.
This can be expressed as
{\color{blue}
g\left(x + fractionReduce(Z,N)
\right)} =
f \left({\color{red}b} \cdot
\left(x + fractionReduce(Z,N)
\right)\right) =
f \left({\color{red}b} x +
\left({\color{red}b} \cdot
fractionReduce(Z,N) \right)\right).
With the given period
\color{orange}{fractionReduce(Z1,N1)}
of f try to choose {\color{red}b} such that
\color{orange}{
{\color{red}b} \cdot fractionReduce(Z,N)
= fractionReduce(Z1,N1)}
holds.
Therefore
f \left({\color{red}b} x +
\left(\color{red}b \cdot \color{orange}{
fractionReduce(Z,N)}\right)\right) =
f \left({\color{red}b} x +
\color{orange}{fractionReduce(Z1,N1)}
\right) =
f \left({\color{red}b} x \right)
={\color{blue}g(x)}.
To get {\color{red}b} solve the equation
\color{orange}{
{\color{red}b} \cdot fractionReduce(Z,N)
= fractionReduce(Z1,N1)}
for {\color{red}b} and find
\displaystyle
{\color{red}b} =
fractionReduce(N*Z1,Z*N1).