Solve a linear congruential algorithm problem

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If we take the linear congruential algorithm with an additive component of 0, namely:

Xn+1 = (aXn) mod m

Then it can be shown that if m is prime, and if a given value of a produces the maximum period of m-1, then a^k will also produce the maximum period, provided that k is less than m and that k and m-1 are relatively prime. Demonstrate this by using X0 = 1 and m = 31 and producing the sequences for a^k = 3 and 32, respectively.

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