Find the LCD 10xy^2 , 15x^2y^3

Since ${\displaystyle 10x{y}^2,15x^2{y}^3}$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part ${\displaystyle 10,15}$ then find LCM for the variable part ${\displaystyle x^1,y^2,x^2,y^3}$.

The LCM is the smallest positive number that all of the numbers divide into evenly.

${\displaystyle 10}$ has factors of ${\displaystyle 2}$ and ${\displaystyle 5}$.

${\displaystyle 15}$ has factors of ${\displaystyle 3}$ and ${\displaystyle 5}$.

The LCM of ${\displaystyle 10,15}$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

The factor for ${\displaystyle x^1}$ is ${\displaystyle x}$ itself.

The factors for ${\displaystyle y^2}$ are ${\displaystyle y\cdot y}$, which is ${\displaystyle y}$ multiplied by each other ${\displaystyle 2}$ times.

The factors for ${\displaystyle x^2}$ are ${\displaystyle x\cdot x}$, which is ${\displaystyle x}$ multiplied by each other ${\displaystyle 2}$ times.

The factors for ${\displaystyle y^3}$ are ${\displaystyle y\cdot y\cdot y}$, which is ${\displaystyle y}$ multiplied by each other ${\displaystyle 3}$ times.

The LCM of ${\displaystyle x^1,y^2,x^2,y^3}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.