Find the greatest common factor. \[ 200 p^{3} m^{3}-100 p^{2} m^{3}+40 m^{3} \]

To find the greatest common factor (GCF) of the expression \( 200 p^{3} m^{3} - 100 p^{2} m^{3} + 40 m^{3} \), we can start by identifying the coefficients and the variables in each term. 1. **Coefficients**: The coefficients are 200, -100, and 40. The GCF of these numbers is 20. 2. **Variable Factors**: All three terms include \( m^{3} \) as a common factor. The first term has \( p^{3} \), the second has \( p^{2} \), and the third has no \( p \). Thus, the lowest power of \( p \) that can be factored out is \( p^{0} \) (which is 1), as \( m^{3} \) is common to all. Putting it all together: The GCF of the entire expression is \( 20 m^{3} \). Now, we can factor the GCF out of the expression: \[ 200 p^{3} m^{3} - 100 p^{2} m^{3} + 40 m^{3} = 20 m^{3} (10 p^{3} - 5 p^{2} + 2) \] So the GCF of \( 200 p^{3} m^{3} - 100 p^{2} m^{3} + 40 m^{3} \) is \( 20 m^{3} \).