Compute the following expression in 𝔽67.
[15] − [22]
Divided by
[8] ⋅ [14]

To solve the expression \(\frac{[15] - [22]}{[8] \cdot [14]}\) in the finite field \(\mathbb{F}_{67}\), we must follow these steps:

1. **Subtraction in \(\mathbb{F}_{67}\):**
   \[
   [15] - [22] \equiv 15 - 22 \equiv -7 \equiv 60 \pmod{67}
   \]
   Here, \(-7\) is equivalent to \(60\) in \(\mathbb{F}_{67}\) because adding 67 to -7 gives us 60.

2. **Multiplication in \(\mathbb{F}_{67}\):**
   \[
   [8] \cdot [14] \equiv 8 \times 14 \equiv 112 \pmod{67}
   \]
   To find \(112 \mod 67\):
   \[
   112 \div 67 = 1 \quad \text{remainder: } 45
   \]
   Thus, \(112 \equiv 45 \pmod{67}\).

3. **Finding the Inverse of \([45]\) in \(\mathbb{F}_{67}\):**
   To compute the division, we need to find the multiplicative inverse of \([45]\) modulo 67. We can use the Extended Euclidean Algorithm to find this inverse.

   Applying the Extended Euclidean Algorithm to \(67\) and \(45\):

   \[
   67 = 1 \times 45 + 22
   \]
   \[
   45 = 2 \times 22 + 1
   \]
   \[
   22 = 22 \times 1 + 0
   \]

   Back-substitution gives us:
   \[
   1 = 45 - 2 \times 22
   \]
   \[
   1 = 45 - 2 \times (67 - 1 \times 45)
   \]
   \[
   1 = 3 \times 45 - 2 \times 67
   \]

   Therefore, the inverse of \(45\) modulo \(67\) is \(3\).

4. **Division in \(\mathbb{F}_{67}\):**
   \[
   \frac{60}{45} \equiv 60 \times 3 \pmod{67}
   \]
   \[
   60 \times 3 = 180
   \]

   To find \(180 \mod 67\):
   \[
   180 \div 67 = 2 \quad \text{remainder: } 46
   \]
   Thus, \(180 \equiv 46 \pmod{67}\).

The final result is \([46]\) in \(\mathbb{F}_{67}\).

If you have any questions or need further details, feel free to ask!

Here are some related questions you might find interesting:
1. How do you find the inverse of a number in a finite field?
2. What is the significance of finite fields in cryptography?
3. How do operations in finite fields differ from those in real numbers?
4. What are some applications of finite fields in computer science?
5. Can you explain the process of the Extended Euclidean Algorithm?
6. How do you perform modular arithmetic efficiently?
7. What are the properties of finite fields?
8. How is subtraction defined in a finite field?

**Tip:** In modular arithmetic, always ensure your results are within the modulus range by adding or subtracting the modulus to adjust negative values or large numbers.

Learn how to compute the expression [15] - [22] divided by [8] multiplied by [14] in the finite field F67 using modular arithmetic and the Extended Euclidean Algorithm. This problem covers operations in finite fields and modular inverses.

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