To find the greatest common divisor (GCD) of 1000 and 5040 using the Euclidean algorithm, we proceed as follows:

1. Apply the Euclidean algorithm:
   • Divide the larger number by the smaller number and find the remainder.
   • Replace the larger number with the smaller number, and the smaller number with the remainder.
   • Repeat this process until the remainder is 0. The last non-zero remainder is the GCD.

Let's perform these steps:

$5040 \div 1000 = 5 \quad \text{(quotient)} \quad \text{remainder} = 5040 - 5 \times 1000 = 5040 - 5000 = 40$

So, we now replace 5040 with 1000 and 1000 with 40.

$1000 \div 40 = 25 \quad \text{(quotient)} \quad \text{remainder} = 1000 - 25 \times 40 = 1000 - 1000 = 0$

Since the remainder is now 0, the last non-zero remainder is 40.

Therefore, GCD(1000, 5040) = 40.

Would you like more details or have any questions?

Here are 5 related questions:

1. How does the Euclidean algorithm relate to the concept of divisibility?
2. Can you find the GCD of 12345 and 67890 using the Euclidean algorithm?
3. What is the significance of finding the GCD in number theory?
4. How does the Euclidean algorithm apply to polynomials?
5. Can you explain the relationship between GCD and the Least Common Multiple (LCM)?

Tip: The Euclidean algorithm is efficient even for very large numbers, making it ideal for computations involving large integers.