Final answer:
In Z/24, 5⁻¹ is equivalent to 5^22. So, 5⁻¹ = 1 (mod 24). Using this result, we can solve the equation 5x + 4 = 1 to find x = -3 (mod 24). However, the equation 6x + 4 = 1 does not have a solution in Z/24.
Step-by-step explanation:
To find 5⁻¹ in Z/24, we can use the concept of modular multiplicative inverse. In Z/24, every number has an inverse if it is coprime with 24. Since 5 is coprime with 24 (gcd(5, 24) = 1), it has an inverse. To find the inverse, we can use the Extended Euclidean Algorithm, which tells us that 5⁻¹ is 5^22 in Z/24. Therefore, 5⁻¹ = 1 (mod 24).
Using the result above, we can solve the equation 5x + 4 = 1 in Z/24. Subtracting 4 from both sides, we get 5x = -3. Dividing both sides by 5, we find that x = -3 * 5⁻¹ = -3 * 1 = -3 (mod 24). So the solution to the equation is x = -3 (mod 24).
For the equation 6x + 4 = 1 in Z/24, we can check if it has a solution by finding the modular multiplicative inverse of 6. However, since gcd(6, 24) = 6, 6 does not have an inverse in Z/24. Therefore, the equation 6x + 4 = 1 does not have a solution in Z/24.