Find the GCF and LCM of 2^3 × 5 × 7^2 × 11 and 2^5 × 3 × 5^2 × 11^2 × 13.

To find the Greatest Common Factor (GCF) and Least Common Multiple (LCM) of the given numbers, we first need to factorize them.

Given numbers:
2^3 × 5 × 7^2 × 11 = 8 × 5 × 49 × 11 = 17640
2^5 × 3 × 5^2 × 11^2 × 13 = 32 × 3 × 25 × 121 × 13 = 254100

Now, we can find the GCF and LCM of 17640 and 254100.

GCF:
Prime factorization of 17640: 2^3 × 3 × 5 × 7^2 × 11
Prime factorization of 254100: 2^2 × 3 × 5^2 × 11^2 × 13

GCF is the product of the common prime factors raised to the lowest power:
GCF = 2^2 × 3 × 5 × 11 = 660

LCM:
LCM is the product of all prime factors raised to the highest power:
LCM = 2^5 × 3 × 5^2 × 7^2 × 11^2 × 13 = 600600

Therefore, the GCF of 17640 and 254100 is 660, and the LCM is 600600.

GCF (Greatest Common Factor):

- Prime factorize both numbers:
- 2^3 × 5 × 7^2 × 11 = 2^3 × 5 × 7^2 × 11
- 2^5 × 3 × 5^2 × 11^2 × 13 = 2^5 × 3 × 5^2 × 11^2 × 13

- Identify common prime factors and their lowest exponents:
- 2: min(3, 5) = 3
- 5: min(1, 2) = 1
- 11: min(2, 2) = 2

- Multiply the common prime factors with their lowest exponents:
- GCF = 2^3 × 5 × 11^2 = 2420

LCM (Least Common Multiple):

- Find the product of all unique prime factors with their highest exponents:
- 2: max(3, 5) = 5
- 3: 1
- 5: max(1, 2) = 2
- 7: 2
- 11: max(2, 2) = 2
- 13: 1

- Multiply the unique prime factors with their highest exponents:
- LCM = 2^5 × 3 × 5^2 × 7^2 × 11^2 × 13 = 53,687,100