In this exercise we consider the multiplicative group Z
∗
31.
1. Determine the order of the element 3

To determine the order of the element \(3\) in the multiplicative group \(\mathbb{Z}_{31}^*\), we follow these steps:

### Step 1: Understand the group \(\mathbb{Z}_{31}^*\)
- The group \(\mathbb{Z}_{31}^*\) consists of the integers from 1 to 30 under multiplication modulo 31.
- Since 31 is a prime number, \(\mathbb{Z}_{31}^*\) is a cyclic group of order \(30\) (because the order of \(\mathbb{Z}_p^*\) is \(p-1\) for a prime \(p\)).

### Step 2: Definition of the order of an element
- The **order** of an element \(a\) in a group is the smallest positive integer \(k\) such that \(a^k \equiv 1 \ (\text{mod} \ 31)\).
- We need to find the smallest \(k\) for which \(3^k \equiv 1 \ (\text{mod} \ 31)\).

### Step 3: Calculate powers of \(3\) modulo \(31\)
We compute successive powers of 3 modulo 31 until we reach 1:

\[
\begin{aligned}
3^1 &\equiv 3 \ (\text{mod} \ 31), \\
3^2 &\equiv 3 \times 3 = 9 \ (\text{mod} \ 31), \\
3^3 &\equiv 3 \times 9 = 27 \ (\text{mod} \ 31), \\
3^4 &\equiv 3 \times 27 = 81 \equiv 19 \ (\text{mod} \ 31), \\
3^5 &\equiv 3 \times 19 = 57 \equiv 26 \ (\text{mod} \ 31), \\
3^6 &\equiv 3 \times 26 = 78 \equiv 16 \ (\text{mod} \ 31), \\
3^7 &\equiv 3 \times 16 = 48 \equiv 17 \ (\text{mod} \ 31), \\
3^8 &\equiv 3 \times 17 = 51 \equiv 20 \ (\text{mod} \ 31), \\
3^9 &\equiv 3 \times 20 = 60 \equiv 29 \ (\text{mod} \ 31), \\
3^{10} &\equiv 3 \times 29 = 87 \equiv 25 \ (\text{mod} \ 31), \\
3^{11} &\equiv 3 \times 25 = 75 \equiv 13 \ (\text{mod} \ 31), \\
3^{12} &\equiv 3 \times 13 = 39 \equiv 8 \ (\text{mod} \ 31), \\
3^{13} &\equiv 3 \times 8 = 24 \ (\text{mod} \ 31), \\
3^{14} &\equiv 3 \times 24 = 72 \equiv 10 \ (\text{mod} \ 31), \\
3^{15} &\equiv 3 \times 10 = 30 \ (\text{mod} \ 31), \\
3^{16} &\equiv 3 \times 30 = 90 \equiv 28 \ (\text{mod} \ 31), \\
3^{17} &\equiv 3 \times 28 = 84 \equiv 22 \ (\text{mod} \ 31), \\
3^{18} &\equiv 3 \times 22 = 66 \equiv 4 \ (\text{mod} \ 31), \\
3^{19} &\equiv 3 \times 4 = 12 \ (\text{mod} \ 31), \\
3^{20} &\equiv 3 \times 12 = 36 \equiv 5 \ (\text{mod} \ 31), \\
3^{21} &\equiv 3 \times 5 = 15 \ (\text{mod} \ 31), \\
3^{22} &\equiv 3 \times 15 = 45 \equiv 14 \ (\text{mod} \ 31), \\
3^{23} &\equiv 3 \times 14 = 42 \equiv 11 \ (\text{mod} \ 31), \\
3^{24} &\equiv 3 \times 11 = 33 \equiv 2 \ (\text{mod} \ 31), \\
3^{25} &\equiv 3 \times 2 = 6 \ (\text{mod} \ 31), \\
3^{26} &\equiv 3 \times 6 = 18 \ (\text{mod} \ 31), \\
3^{27} &\equiv 3 \times 18 = 54 \equiv 23 \ (\text{mod} \ 31), \\
3^{28} &\equiv 3 \times 23 = 69 \equiv 7 \ (\text{mod} \ 31), \\
3^{29} &\equiv 3 \times 7 = 21 \ (\text{mod} \ 31), \\
3^{30} &\equiv 3 \times 21 = 63 \equiv 1 \ (\text{mod} \ 31).
\end{aligned}
\]

### Step 4: Conclusion
The order of the element \(3\) in \(\mathbb{Z}_{31}^*\) is \(30\) since \(3^{30} \equiv 1 \ (\text{mod} \ 31)\), and this is the smallest such exponent.

---

Would you like to know more about cyclic groups, or do you have any other questions on this topic?

### Further Questions:
1. How can we determine if \(\mathbb{Z}_{31}^*\) is cyclic?
2. What is the significance of the order of an element in group theory?
3. How do we find the generator of a cyclic group like \(\mathbb{Z}_{31}^*\)?
4. What are the possible orders of elements in \(\mathbb{Z}_{31}^*\)?
5. Can the order of an element be related to the structure of the group?

### Tip:
When calculating the order of an element, if the group order is large, it is helpful to compute powers incrementally, paying attention to any repeated values.

Explore how to determine the order of the element 3 in the multiplicative group Z*31 using modular arithmetic and cyclic group theory. Follow a step-by-step calculation to find the smallest exponent for which 3 raised to that power equals 1 modulo 31.

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