Question

GCF of 315,450
GCF of 60,792
GCF of 585,567
GCF of 75, 150

Answer 1

To find the Greatest Common Factor (GCF) of the given pairs of numbers, we can use the prime factorization method or the Euclidean algorithm. Here, I will calculate the GCF for each pair of numbers you provided.

### 1. GCF of 315 and 450:
- Prime factorization of 315:
- 315 = 3 × 3 × 5 × 7 = \(3^2 \times 5^1 \times 7^1\)

- Prime factorization of 450:
- 450 = 2 × 3 × 3 × 5 × 5 = \(2^1 \times 3^2 \times 5^2\)

- GCF:
- Take the lowest power of each common prime factor:
- GCF = \(3^2 \times 5^1 = 9 \times 5 = 45\)

### GCF(315, 450) = 45

### 2. GCF of 60 and 792:
- Prime factorization of 60:
- 60 = 2 × 2 × 3 × 5 = \(2^2 \times 3^1 \times 5^1\)

- Prime factorization of 792:
- 792 = 2 × 2 × 2 × 3 × 11 = \(2^3 \times 3^1 \times 11^1\)

- GCF:
- GCF = \(2^2 \times 3^1 = 4 \times 3 = 12\)

### GCF(60, 792) = 12

### 3. GCF of 585 and 567:
- Prime factorization of 585:
- 585 = 3 × 5 × 39 = 3 × 5 × 3 × 13 = \(3^2 \times 5^1 \times 13^1\)

- Prime factorization of 567:
- 567 = 3 × 189 = 3 × 3 × 63 = 3 × 3 × 3 × 21 = \(3^4 \times 7^1\)

- GCF:
- GCF = \(3^2 = 9\)

### GCF(585, 567) = 9

### 4. GCF of 75 and 150:
- Prime factorization of 75:
- 75 = 3 × 5 × 5 = \(3^1 \times 5^2\)

- Prime factorization of 150:
- 150 = 2 × 3 × 5 × 5 = \(2^1 \times 3^1 \times 5^2\)

- GCF:
- GCF = \(3^1 \times 5^2 = 3 \times 25 = 75\)

### GCF(75, 150) = 75

### Summary of GCFs:
- GCF(315, 450) = 45
- GCF(60, 792) = 12
- GCF(585, 567) = 9
- GCF(75, 150) = 75