Random number generator RNG and periodic outputs

The problem with such generators is that their outputs are periodic; they will start to repeat themselves with a period that is at most $M$. If however the parameters $a$ and $c$ are badly chosen, the period may be even shorter.

Consider the following example $$\begin{equation*} N_i=(6N_{i-1}+7) \mathrm{MOD} (5), \end{equation*}$$ with a seed $N_0=2$. This generator produces the sequence $4,1,3,0,2,4,1,3,0,2,...\dots$, i.e., a sequence with period $5$. However, increasing $M$ may not guarantee a larger period as the following example shows $$\begin{equation*} N_i=(27N_{i-1}+11) \mathrm{MOD} (54), \end{equation*}$$ which still, with $N_0=2$, results in $11,38,11,38,11,38,\dots$, a period of just $2$.