The image contains a mathematical prompt that reads:

1. Use the Euclidean algorithm to compute the greatest common divisor (GCD) of $63 + 5$ and $3 + 8$, where $n$ is some positive integer.

The goal of this problem is to apply the Euclidean algorithm to find the GCD of two numbers. Let's break it down:

The two numbers are: $63 + 5 = 68$ $3 + 8 = 11$

Now, using the Euclidean algorithm, we compute the GCD of 68 and 11.

Steps:

1. Divide 68 by 11: $68 \div 11 = 6 \text{ remainder } 2$ So, $68 = 11 \times 6 + 2$.

2. Now, divide 11 by the remainder 2: $11 \div 2 = 5 \text{ remainder } 1$ So, $11 = 2 \times 5 + 1$.

3. Finally, divide 2 by the remainder 1: $2 \div 1 = 2 \text{ remainder } 0$ Since the remainder is 0, the GCD is 1.

Thus, the greatest common divisor (GCD) of 68 and 11 is 1.

Would you like more details or further questions on this? Here are 5 related questions:

1. Can you apply the Euclidean algorithm to other pairs of numbers, like 84 and 18?
2. What are the practical applications of finding the GCD in number theory?
3. How would the Euclidean algorithm change if we wanted the least common multiple (LCM)?
4. Can the Euclidean algorithm be extended to find GCDs of more than two numbers?
5. How does the GCD relate to the concept of relatively prime numbers?

Tip: The Euclidean algorithm is one of the most efficient ways to compute the GCD of two numbers, and it can be adapted for larger numbers easily!