To compute the expression in the finite field $\mathbb{F}_{67}$, we need to perform the operations modulo 67. The expression is:

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

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

$15 - 22 = -7$ Since we're working in $\mathbb{F}_{67}$, we need to find the equivalent value 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, find $112 \mod 67$: $112 \mod 67 = 45$ So, $[8] \cdot [14] \equiv 45 \mod 67$.

Step 3: Compute the multiplicative inverse of 45 in $\mathbb{F}_{67}$

To find $\frac{60}{45}$ in $\mathbb{F}_{67}$, we need the multiplicative inverse of 45 modulo 67. This means finding $x$ such that: $45x \equiv 1 \mod 67$ Using the Extended Euclidean Algorithm, we find that the inverse of 45 modulo 67 is 37.

Step 4: Multiply the result of Step 1 by the inverse found in Step 3

$\frac{60}{45} \equiv 60 \times 37 \mod 67$ $60 \times 37 = 2220$ Now, find $2220 \mod 67$: $2220 \mod 67 = 6$

Final Answer:

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

Would you like more details or have any questions?

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Related Questions:

1. How do you find the multiplicative inverse of a number in $\mathbb{F}_p$?
2. What is the general method for solving division in finite fields?
3. How does the Extended Euclidean Algorithm work in finding inverses?
4. Can you explain the properties of arithmetic in $\mathbb{F}_p$?
5. What would happen if the divisor had no inverse in $\mathbb{F}_{67}$?
6. How do you perform addition and subtraction in finite fields?
7. What is modular arithmetic, and why is it important in finite fields?
8. Can you give an example of a real-world application where finite fields are used?

Tip: When working in a finite field, always ensure the divisor has an inverse before dividing.