What is the greatest common factor of 60x^4y^7, 45x^5y^5, and 75x^3y?

The greatest common factor (GCF) is the largest expression that divides each term in a set of terms without leaving a remainder. To determine the GCF of the expressions 60x⁴y⁷, 45x⁵y⁵, and 75x³y, we need to find the GCF for both the numerical coefficients and the variable parts separately.

1. **Finding the GCF of the coefficients (60, 45, and 75):**

First, we can find the prime factorization of each number:

• 60 = 2 × 2 × 3 × 5 = 2² × 3¹ × 5¹
• 45 = 3 × 3 × 5 = 3² × 5¹
• 75 = 3 × 5 × 5 = 3¹ × 5²

Next, we take the lowest power of all prime factors found in these factorizations:

• For 2, the lowest power is 0 (not present in 45 or 75)
• For 3, the lowest power is 1
• For 5, the lowest power is 1

So, the GCF of the coefficients is:

GCF = 3¹ × 5¹ = 15

2. **Finding the GCF of the variable parts (x and y):**

For the variable part, we need to find the lowest power of each variable present:

• x appears in 60x⁴y⁷ as x⁴, in 45x⁵y⁵ as x⁵, and in 75x³y as x³. The lowest power is 3.
• y appears in 60x⁴y⁷ as y⁷, in 45x⁵y⁵ as y⁵, and in 75x³y as y¹. The lowest power is 1.

Putting it all together, the GCF of the variable parts is x³y¹.

3. **Final GCF:**

Now we combine the GCF of the coefficients and the variables:

GCF = 15x³y

Therefore, the greatest common factor of 60x⁴y⁷, 45x⁵y⁵, and 75x³y is 15x³y.