  A $K$-Fibonacci sequence is a binary recurrence sequence where $F_0=0$,
$F_1=1$, and $F_n=K\cdot F_{n-1}+F_{n-2}$. These sequences are known to be
periodic modulo every positive integer greater than $1$. If the length of one
shortest period of a $K$-Fibonacci sequence modulo a positive integer is equal
to the modulus, then that positive integer is called a $\textit{fixed point}$.
This paper determines the fixed points of $K$-Fibonacci sequences according to
the factorization of $K^2+4$ and concludes that if this process is iterated,
then every modulus greater than $3$ eventually terminates at a fixed point.
