Find the greatest common factor. \( 10 \mathrm{~m}^{3}, 2 \mathrm{~m}^{2} \) Write your answer as a constant times a product of single variables raised to exponents.

To find the greatest common factor (GCF) of \( 10 \mathrm{~m}^{3} \) and \( 2 \mathrm{~m}^{2} \), we start by breaking down the expressions into their components. The coefficients \( 10 \) and \( 2 \) have a GCF of \( 2 \). For the variable part, \( m^3 \) and \( m^2 \), the GCF is determined by the lowest exponent, which is \( m^2 \). Thus, the GCF is: \[ 2 \mathrm{~m}^{2} \] If we wanted to express it in the specified format, it would be: \[ 2 \cdot m^{2} \] --- While the concept of the greatest common factor is straightforward, it has roots in ancient mathematics. The ancient Greeks, particularly Euclid, contributed significantly to this area, developing algorithms to find common divisors long before modern mathematics fully formed. It was their systematic approach that laid the groundwork for what we now refer to as the Euclidean algorithm. In real-life situations, understanding factors can help simplify problems, especially in areas like construction or cooking. For instance, when scaling recipes or determining the size of materials, knowing the GCF allows one to minimize waste and ensure the best fit between parts or ingredients, making the processes far more efficient!