To compute the expression in the finite field $\mathbb{F}_{67}$, we will follow these steps:

Expression:

[ \frac{[15] - [22]}{[8] \cdot [14]} ]

Step 1: Compute $[15] - [22]$ in $\mathbb{F}_{67}$

$15 - 22 = -7$ In $\mathbb{F}_{67}$, we need to reduce this modulo 67: $-7 \equiv 60 \mod 67$ So, $[15] - [22] \equiv 60 \mod 67$.

Step 2: Compute $[8] \cdot [14]$ in $\mathbb{F}_{67}$

$8 \times 14 = 112$ Now, reduce this modulo 67: $112 \mod 67 = 45$ So, $[8] \cdot [14] \equiv 45 \mod 67$.

Step 3: Compute the final result in $\mathbb{F}_{67}$

Now, we need to find the expression $\frac{60}{45}$ modulo 67. This can be expressed as: $\frac{60}{45} \equiv 60 \times 45^{-1} \mod 67$ where $45^{-1}$ is the multiplicative inverse of 45 modulo 67.

Step 4: Find the inverse of 45 modulo 67

Using the Extended Euclidean Algorithm or trial and error, we find that the inverse of 45 modulo 67 is 37.

Step 5: Multiply and reduce the result modulo 67

$60 \times 37 = 2220$ Now reduce this modulo 67: $2220 \mod 67 = 6$

Final Answer:

[ \frac{[15] - [22]}{[8] \cdot [14]} \equiv 6 \mod 67 ]

The final answer is $6$ in $\mathbb{F}_{67}$.

Would you like more details or have any questions?

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Related Questions:

1. How do you find the multiplicative inverse in a finite field?
2. Why is modular arithmetic important in number theory?
3. How can you verify your computations in finite fields?
4. What are common applications of finite fields in cryptography?
5. How does the Extended Euclidean Algorithm work?
6. Can you compute similar expressions in other finite fields?
7. What happens if there is no inverse for the divisor in a finite field?
8. How are finite fields used in error detection and correction?

Tip: Understanding how to find inverses in a finite field is crucial for solving division problems in modular arithmetic.