 Solution

A linear congruential generator starts with an integer seed $S_0\in \{0,\dots ,M-1\}$ and iteratesThe outputs lie in $[0,1)$ and are used as pseudorandom numbers approximating a uniform distribution. They are deterministic once the seed is fixed. The largest possible period is $M$, since there are only $M$ states; good period alone does not establish independence or good multivariate uniformity.

The full-period criterion for a mixed congruential generator says that period $M$ for every seed occurs exactly whenFor $M=12^k=2^{2k}{3}^k$, these conditions reduce toThe multiplier must satisfy both $a\equiv 1(\mathrm{mod}3)$ and $a\equiv 1(\mathrm{mod}4)$. A positive shift alone is insufficient: for example, a shift divisible by three cannot give period $M$ when the multiplier is one modulo three. If the required coprime shift has already been chosen, the condition on the multiplier alone is $a=1+12r$ for an integer $r$.