Consider the multiplicative group z31. Calculate your order of the element 3

To find the order of the element \(3\) in the multiplicative group \( \mathbb{Z}_{31}^* \), we need to determine the smallest positive integer \( k \) such that:

\[
3^k \equiv 1 \ (\text{mod} \ 31)
\]

Here's how we proceed:

1. **Check the order of the group**: The order of the multiplicative group \( \mathbb{Z}_{31}^* \) is \( \phi(31) = 31 - 1 = 30 \), where \( \phi \) is Euler's totient function. Therefore, the possible values of \( k \) are divisors of 30.

2. **Test each divisor**: The divisors of 30 are \(1, 2, 3, 5, 6, 10, 15, 30\). We need to find the smallest \( k \) for which \( 3^k \equiv 1 \ (\text{mod} \ 31) \).

   - \( 3^1 \equiv 3 \ (\text{mod} \ 31) \)
   - \( 3^2 = 9 \equiv 9 \ (\text{mod} \ 31) \)
   - \( 3^3 = 27 \equiv 27 \ (\text{mod} \ 31) \)
   - \( 3^5 = 243 \equiv 26 \ (\text{mod} \ 31) \)
   - \( 3^6 = 3^3 \times 3^3 = 27 \times 27 = 729 \equiv 16 \ (\text{mod} \ 31) \)
   - \( 3^{10} = (3^5)^2 = 26^2 = 676 \equiv 26 \ (\text{mod} \ 31) \)
   - \( 3^{15} = 3^{10} \times 3^5 = 26 \times 26 = 676 \equiv 26 \ (\text{mod} \ 31) \)
   - \( 3^{30} = (3^{15})^2 = 26^2 \equiv 1 \ (\text{mod} \ 31) \)

So, \( k = 30 \) is the smallest positive integer for which \( 3^{30} \equiv 1 \ (\text{mod} \ 31) \).

Thus, the order of the element \( 3 \) in \( \mathbb{Z}_{31}^* \) is **30**.

Would you like more details or have any questions about this? Here are some additional questions related to the topic:

1. What is the general method for finding the order of an element in a multiplicative group?
2. How does Euler's totient function \( \phi(n) \) relate to the group \( \mathbb{Z}_n^* \)?
3. Can the order of any element in \( \mathbb{Z}_p^* \) (where \( p \) is prime) ever be greater than \( \phi(p) \)?
4. What is the significance of the order of an element in group theory?
5. How can we verify that \( \mathbb{Z}_{31}^* \) is a cyclic group?

**Tip**: In a multiplicative group \( \mathbb{Z}_p^* \) for a prime \( p \), the order of any element always divides \( p-1 \).

Learn how to calculate the order of the element 3 in the multiplicative group Z_31*. This problem involves modular arithmetic, Euler's totient function, and properties of cyclic groups. Suitable for undergraduate students studying group theory.

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