Find the GCF and LCM of: 15, 39, 105 2^3 × 3^2 × 5 × 7 and 2^2 × 3^3 × 7^2 × 11

To find the GCF and LCM of the numbers given, we first need to find the prime factorization of each number.

For 15:
15 = 3 * 5
Prime factorization: 3 * 5

For 39:
39 = 3 * 13
Prime factorization: 3 * 13

For 105:
105 = 3 * 5 * 7
Prime factorization: 3 * 5 * 7

Now, we can find the GCF by taking the product of the common prime factors raised to the lowest power:
GCF = 3 * 5 = 15

To find the LCM, we take the product of all prime factors raised to the highest power:
LCM = 3 * 5 * 7 * 13 = 1365

For the second set of numbers:

2^3 * 3^2 * 5 * 7 = 8 * 9 * 5 * 7 = 2520
Prime factorization: 2^3 * 3^2 * 5 * 7

2^2 * 3^3 * 7^2 * 11 = 4 * 27 * 49 * 11 = 52488
Prime factorization: 2^2 * 3^3 * 7^2 * 11

Now, we can find the GCF by taking the product of the common prime factors raised to the lowest power:
GCF = 2^2 * 3^2 * 7 = 252

To find the LCM, we take the product of all prime factors raised to the highest power:
LCM = 2^3 * 3^3 * 5 * 7^2 * 11 = 55440

Therefore, the GCF of 15, 39, and 105 is 15, and the LCM is 1365. The GCF of 2520 and 52488 is 252, and the LCM is 55440.

1. GCF and LCM of 15, 39, 105

- GCF:
- Factorize each number:
- 15 = 3 × 5
- 39 = 3 × 13
- 105 = 3 × 5 × 7
- The common factors are 3 and 5.
- Therefore, GCF = 3 × 5 = 15
- LCM:
- List all the prime factors: 3, 5, 7, 13
- Include the highest power of each factor: 3^1, 5^1, 7^1, 13^1
- Multiply these factors:
- LCM = 3^1 × 5^1 × 7^1 × 13^1 = 390

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

- GCF:
- Extract the common prime factors: 2^2, 3^2, and 7^1
- Multiply these factors:
- GCF = 2^2 × 3^2 × 7^1 = 126
- LCM:
- List all the prime factors: 2, 3, 5, 7, 11
- Include the highest power of each factor: 2^3, 3^3, 5^1, 7^2, 11^1
- Multiply these factors:
- LCM = 2^3 × 3^3 × 5^1 × 7^2 × 11^1 = 30870