1011. Pseudo-Random Numbers

Computers normally cannot generate really random numbers, but frequently are used to generate sequences of pseudo-random numbers. These are generated by some algorithm, but appear for all practical purposes to be really random. Random numbers are used in many applications, including simulation.

A common pseudo-random number generation technique is called the linear congruential method. If the last pseudo-random number generated was $L$, then the next number is generated by evaluating $(Z\cdot L+I)modM$, where $Z$ is a constant multiplier, $I$ is a constant increment, and $M$ is a constant modulus. For example, suppose $Z$ is $7$, $I$ is $5$, and $M$ is $12$. If the first random number (usually called the seed) is $4$, then we can determine the next few pseudo-random numbers are follows:

As you can see, the sequence of pseudo-random numbers generated by this technique repeats after six numbers. It should be clear that the longest sequence that can be generated using this technique is limited by the modulus, $M$.

In this problem you will be given sets of values for $Z,I,M$, and the seed, $L$. Each of these will have no more than four digits. For each such set of values you are to determine the length of the cycle of pseudo-random numbers that will be generated. But be careful: the cycle might not begin with the seed!