To determine the greatest common factor (GCF) of the terms [tex]\( 60 x^4 y^7 \)[/tex], [tex]\( 45 x^5 y^5 \)[/tex], and [tex]\( 75 x^3 y \)[/tex], we will address both the coefficients and the variable parts separately.
### Step-by-Step Approach:
1. Find the GCF of the coefficients:
- The coefficients are [tex]\(60\)[/tex], [tex]\(45\)[/tex], and [tex]\(75\)[/tex].
- To find the GCF of these numbers, we can use their prime factorizations:
[tex]\[
60 = 2^2 \cdot 3 \cdot 5
\][/tex]
[tex]\[
45 = 3^2 \cdot 5
\][/tex]
[tex]\[
75 = 3 \cdot 5^2
\][/tex]
- The common prime factors across all three numbers are [tex]\(3\)[/tex] and [tex]\(5\)[/tex].
- Taking the lowest power of each common prime factor:
[tex]\[
GCF = 3^1 \cdot 5^1 = 15
\][/tex]
2. Find the GCF of the variable powers for [tex]\(x\)[/tex]:
- The exponents of [tex]\(x\)[/tex] in the terms are [tex]\(4\)[/tex], [tex]\(5\)[/tex], and [tex]\(3\)[/tex].
[tex]\[
GCF(x^4, x^5, x^3) = x^{\min(4, 5, 3)} = x^3
\][/tex]
3. Find the GCF of the variable powers for [tex]\(y\)[/tex]:
- The exponents of [tex]\(y\)[/tex] in the terms are [tex]\(7\)[/tex], [tex]\(5\)[/tex], and [tex]\(1\)[/tex].
[tex]\[
GCF(y^7, y^5, y^1) = y^{\min(7, 5, 1)} = y^1 = y
\][/tex]
4. Combine the GCFs:
- We have determined the GCFs of the coefficients and the variables separately. Now, combine them:
[tex]\[
GCF = 15 \cdot x^3 \cdot y = 15x^3y
\][/tex]
### Conclusion:
The greatest common factor of [tex]\( 60 x^4 y^7 \)[/tex], [tex]\( 45 x^5 y^5 \)[/tex], and [tex]\( 75 x^3 y \)[/tex] is:
[tex]\[
\boxed{15 x^3 y}
\][/tex]
