Final answer:
The greatest common divisor (GCD) of the given pair of integers, 2² · 3³ · 5 · 7 · 11² · 13 and 2⁹ · 3⁸ · 11¹ · 17, is 58,464.
Step-by-step explanation:
The greatest common divisor (GCD) is the largest number that divides both given integers without leaving a remainder. To find the GCD of 2² · 3³ · 5 · 7 · 11² · 13 and 2⁹ · 3⁸ · 11¹ · 17, we need to consider the common factors between the two numbers.
Let's break down both numbers into their prime factorizations:
- 2² · 3³ · 5 · 7 · 11² · 13 = 2 · 2 · 3 · 3 · 3 · 5 · 7 · 11 · 11 · 13
- 2⁹ · 3⁸ · 11¹ · 17 = 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2 · 3 · 3 · 3 · 3 · 3 · 3 · 3 · 11 · 17
To find the GCD, we need to identify the highest power of each common prime factor. In this case, the highest power of 2 is 2⁷, the highest power of 3 is 3³, and the highest power of 11 is 11¹. Therefore, the GCD of the given pair of integers is 2⁷ · 3³ · 11¹ = 2,048 · 27 · 11 = 58,464.