We can determine the greatest common divisor (gcd) of 2516 and 1125 using the Euclidean algorithm. The steps are as follows:

1. Divide the larger number by the smaller number and find the remainder: $2516 \div 1125 = 2 \text{ remainder } 266$ So, $2516 = 1125 \times 2 + 266$.

2. Now, divide 1125 by the remainder (266): $1125 \div 266 = 4 \text{ remainder } 61$ So, $1125 = 266 \times 4 + 61$.

3. Divide 266 by the remainder (61): $266 \div 61 = 4 \text{ remainder } 22$ So, $266 = 61 \times 4 + 22$.

4. Divide 61 by the remainder (22): $61 \div 22 = 2 \text{ remainder } 17$ So, $61 = 22 \times 2 + 17$.

5. Divide 22 by the remainder (17): $22 \div 17 = 1 \text{ remainder } 5$ So, $22 = 17 \times 1 + 5$.

6. Divide 17 by the remainder (5): $17 \div 5 = 3 \text{ remainder } 2$ So, $17 = 5 \times 3 + 2$.

7. Divide 5 by the remainder (2): $5 \div 2 = 2 \text{ remainder } 1$ So, $5 = 2 \times 2 + 1$.

8. Finally, divide 2 by the remainder (1): $2 \div 1 = 2 \text{ remainder } 0$ Since the remainder is now 0, the gcd is the last non-zero remainder, which is 1.

Thus, the greatest common divisor (gcd) of 2516 and 1125 is 1.

Would you like more details or have any other questions?

Here are five related questions:

1. What is the least common multiple (LCM) of 2516 and 1125?
2. How does the Euclidean algorithm work for larger numbers?
3. Can the gcd of two numbers be larger than either of the two numbers?
4. How is the gcd related to prime factorization?
5. What is the gcd of two consecutive Fibonacci numbers?

Tip: The Euclidean algorithm is efficient and works well even for large numbers!