1. **State the problem:** Find the greatest common factor (GCF) of the monomials $35x^3y^2$, $10x^4y$, and $5x^5y^3$. 2. **Recall the formula and rules:** The GCF of monomials is found by taking the GCF of the coefficients and the lowest powers of each variable common to all monomials. 3. **Find the GCF of the coefficients:** The coefficients are 35, 10, and 5. - Factors of 35: $1, 5, 7, 35$ - Factors of 10: $1, 2, 5, 10$ - Factors of 5: $1, 5$ The greatest common factor of the coefficients is $5$. 4. **Find the GCF of the variables:** - For $x$: powers are $3, 4, 5$. The lowest power is $3$, so GCF for $x$ is $x^3$. - For $y$: powers are $2, 1, 3$. The lowest power is $1$, so GCF for $y$ is $y^1 = y$. 5. **Combine the GCF of coefficients and variables:** $$\text{GCF} = 5x^3y$$ 6. **Final answer:** The greatest common factor of $35x^3y^2$, $10x^4y$, and $5x^5y^3$ is **$5x^3y$**.