Answer :

Final answer:

To find the maximum order of any element of U(500), calculate the number of positive integers less than 500 coprime with 500 using Euler's totient function. Then, for each number 'a' coprime with 500, calculate a^n mod 500 until finding an 'n' where a^n is congruent to 1 modulo 500.

Explanation:

The maximum order of any element of U(500) can be determined by finding the smallest positive integer 'n' such that a^n is congruent to 1 modulo 500, where 'a' is an element of U(500).

The set U(500) consists of all the positive integers less than 500 that are coprime with 500.

  1. Calculate the number of positive integers less than 500 that are coprime with 500. This can be done using Euler's totient function, φ(500).
  2. For each number 'a' coprime with 500, calculate a^n modulo 500 for increasing values of 'n' until you find an 'n' such that a^n ≡ 1 (mod 500).
  3. The maximum order of any element of U(500) is the highest 'n' value obtained in the previous step.

For example, U(500) = {1, 3, 7, 9, 11, 13, 17, 19, ...} and

φ(500) = 200.

Taking 'a' = 3 as an example, we can find that 3^40 ≡ 1 (mod 500),

so the maximum order of any element of U(500) is 40.

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