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A linear congruential generator has period $M$ for every seed if and only if $\gcd (c,M)=1$, every prime dividing $M$ divides $a-1$, and $4\mid M$ implies $4\mid a-1$. The coprimality of $c$ prevents the orbit from being trapped in a residue class; the multiplier conditions permit full-period lifting through each prime power. Combining the prime-power periods by the Chinese remainder theorem gives period $M$. For $M=12^k=2^{2k}{3}^k$, the multiplier criterion reduces to $a\equiv 1(\mathrm{mod}12)$, together with $\gcd (c,6)=1$. A positive increment alone is insufficient.

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 $r$.

A pseudorandom number is an output of a deterministic algorithm chosen to imitate samples from a specified probability law. For example, a linear congruential generator returns fractions $S_i/M$ on a finite grid in $[0,1)$. Their intended uniform distribution is an approximation, and their serial dependence must be assessed separately from the generator's period.