In this exercise we consider the multiplicative group Z
∗
31.
1. Determine the order of the element 3.
2. Calculate 3
274 mod 31.
3. Find a solution for x for the equation 3
7x+25 ≡ −12 mod 31.


Let's solve the three parts of this exercise one by one.

### 1. Determine the order of the element \(3\) in \(\mathbb{Z}_{31}^*\)

The order of an element \(a\) in a group \(\mathbb{Z}_{n}^*\) is the smallest positive integer \(k\) such that \(a^k \equiv 1 \mod n\).

We are given \(n = 31\) and \(a = 3\).

To find the order, we need to calculate successive powers of 3 modulo 31 until we get 1.

\[
3^1 \mod 31 = 3
\]
\[
3^2 \mod 31 = 9
\]
\[
3^3 \mod 31 = 27
\]
\[
3^4 \mod 31 = 81 \equiv 19 \mod 31
\]
\[
3^5 \mod 31 = 3 \times 19 = 57 \equiv 26 \mod 31
\]
\[
3^6 \mod 31 = 3 \times 26 = 78 \equiv 16 \mod 31
\]
\[
3^7 \mod 31 = 3 \times 16 = 48 \equiv 17 \mod 31
\]
\[
3^8 \mod 31 = 3 \times 17 = 51 \equiv 20 \mod 31
\]
\[
3^9 \mod 31 = 3 \times 20 = 60 \equiv 29 \mod 31
\]
\[
3^{10} \mod 31 = 3 \times 29 = 87 \equiv 25 \mod 31
\]
\[
3^{11} \mod 31 = 3 \times 25 = 75 \equiv 13 \mod 31
\]
\[
3^{12} \mod 31 = 3 \times 13 = 39 \equiv 8 \mod 31
\]
\[
3^{13} \mod 31 = 3 \times 8 = 24 \mod 31
\]
\[
3^{14} \mod 31 = 3 \times 24 = 72 \equiv 10 \mod 31
\]
\[
3^{15} \mod 31 = 3 \times 10 = 30 \mod 31
\]
\[
3^{16} \mod 31 = 3 \times 30 = 90 \equiv 29 \mod 31
\]
\[
3^{17} \mod 31 = 3 \times 29 = 87 \equiv 25 \mod 31
\]
\[
3^{18} \mod 31 = 3 \times 25 = 75 \equiv 13 \mod 31
\]
\[
3^{19} \mod 31 = 3 \times 13 = 39 \equiv 8 \mod 31
\]
\[
3^{20} \mod 31 = 3 \times 8 = 24 \mod 31
\]
\[
3^{21} \mod 31 = 3 \times 24 = 72 \equiv 10 \mod 31
\]
\[
3^{22} \mod 31 = 3 \times 10 = 30 \mod 31
\]
\[
3^{23} \mod 31 = 3 \times 30 = 90 \equiv 29 \mod 31
\]
\[
3^{24} \mod 31 = 3 \times 29 = 87 \equiv 25 \mod 31
\]
\[
3^{25} \mod 31 = 3 \times 25 = 75 \equiv 13 \mod 31
\]
\[
3^{26} \mod 31 = 3 \times 13 = 39 \equiv 8 \mod 31
\]
\[
3^{27} \mod 31 = 3 \times 8 = 24 \mod 31
\]
\[
3^{28} \mod 31 = 3 \times 24 = 72 \equiv 10 \mod 31
\]
\[
3^{29} \mod 31 = 3 \times 10 = 30 \mod 31
\]
\[
3^{30} \mod 31 = 3 \times 30 = 90 \equiv 1 \mod 31
\]

Thus, the order of \(3\) is **30**.

### 2. Calculate \(3^{274} \mod 31\)

We can use the order we found in the previous part to simplify this calculation using the fact that \(3^{30} \equiv 1 \mod 31\).

First, reduce the exponent modulo 30:
\[
274 \mod 30 = 4
\]
Thus,
\[
3^{274} \equiv 3^4 \mod 31
\]
From the previous calculation:
\[
3^4 \equiv 19 \mod 31
\]
So,
\[
3^{274} \equiv 19 \mod 31
\]

### 3. Find a solution for \(x\) in the equation \(3^{7x+25} \equiv -12 \mod 31\)

First, convert \(-12\) to a positive residue modulo 31:
\[
-12 \equiv 19 \mod 31
\]
We need to solve:
\[
3^{7x+25} \equiv 19 \mod 31
\]
Since \(3^{30} \equiv 1 \mod 31\), \(19\) is equivalent to \(3^4\) mod 31. Therefore, we need:
\[
7x + 25 \equiv 4 \mod 30
\]
Solve for \(x\):
\[
7x \equiv 4 - 25 \mod 30
\]
\[
7x \equiv -21 \equiv 9 \mod 30
\]
Now, we find the inverse of 7 modulo 30. The inverse is the number \(y\) such that:
\[
7y \equiv 1 \mod 30
\]
By trial and error or using the extended Euclidean algorithm, the inverse of 7 modulo 30 is \(13\). Thus:
\[
x \equiv 13 \times 9 \mod 30
\]
\[
x \equiv 117 \equiv 27 \mod 30
\]
Thus, the solution is \(x \equiv 27 \mod 30\).

### Summary
1. The order of 3 in \(\mathbb{Z}_{31}^*\) is **30**.
2. \(3^{274} \equiv 19 \mod 31\).
3. The solution for \(x\) in the equation \(3^{7x+25} \equiv -12 \mod 31\) is \(x \equiv 27 \mod 30\).

Would you like more details or have any questions?

Here are 5 related questions:
1. How do you determine the order of an element in any group?
2. What are the properties of cyclic groups in modular arithmetic?
3. How can the extended Euclidean algorithm help solve linear congruences?
4. What is the significance of reducing exponents in modular arithmetic?
5. How can you use Fermat's Little Theorem in problems like these?

**Tip:** When solving congruences involving modular arithmetic, breaking down the problem using properties like the order of elements can significantly simplify calculations.

Explore solutions to mathematical problems involving the multiplicative group Z*31. Calculate the order of element 3, find 3^274 mod 31, and solve the equation 3^(7x+25) ≡ -12 mod 31 step-by-step with detailed explanations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation