Question

A. What is the maximum period obtainable from the following generator? Xn+1 =(aXn)mod24

b. What should be the value of a ? c. What restrictions are required on the seed?

Answer 1

The maximum period obtainable from the given generator is 24. The value of a can be determined to be 23. The restrictions on the seed, X0, are that it must be a non-negative integer less than b.

Step-by-step explanation:

Based on the given generator formula, Xn+1 = (aXn) mod 24b, the maximum period obtainable is 24. This means that the generator will cycle through all 24 possible values before repeating.

To find the value of a, we need to determine the largest possible value that will not violate any restrictions. Since the generator produces values between 0 and 24, a possible value for a could be 23.

The restrictions required on the seed, X0, are that it must be a non-negative integer less than b.

Answer 2

a. The maximum period obtainable from the given generator is 8.

b. The value of \(a\) should be 3.

c. The seed should be a positive integer less than 8.

Step-by-step explanation:

The generator function is
`\(X_(n+1) = (aX_n) \mod 8\)`. To find the maximum period, we need to determine the order of
`\(a\)` modulo 8. The values of
`\(a\)`that result in the maximum period are those where \(a\) is coprime to 8. Since 3 is the only positive integer less than 8 that is coprime to 8, the optimal value for
`\(a\)` is 3.

To elaborate, the generator's period is determined by the order of
`\(a\)`modulo 8. The order of an integer
`\(a\)` modulo
`\(m\)` is the smallest positive integer
`\(r\)` such that
`\(a^r \equiv 1 \mod m\)`. In this case,
`\(3^1 \equiv 3\), \(3^2 \equiv 1 \mod 8\)`, so the order of 3 modulo 8 is 2. Therefore, the maximum period is
`\(8/2 = 4\)`, and to achieve this,
`\(a\)` must be 3.

Regarding the seed, since the generator is modulo 8, the seed should be a positive integer less than 8 to ensure that the output remains within the desired range. This restriction on the seed guarantees that the generator will produce a repeating sequence with a maximum period of 4 when
`\(a\)` is set to 3.