Find the LCD 1/(3x) , 2/(7x) , (x+2)/(14x)

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

Since ${\displaystyle 3x,7x,14x}$ contain both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part ${\displaystyle 3,7,14}$ then find LCM for the variable part ${\displaystyle x^1,x^1,x^1}$.

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

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Since ${\displaystyle 3}$ has no factors besides ${\displaystyle 1}$ and ${\displaystyle 3}$.

${\displaystyle 3}$ is a prime number

Since ${\displaystyle 7}$ has no factors besides ${\displaystyle 1}$ and ${\displaystyle 7}$.

${\displaystyle 7}$ is a prime number

${\displaystyle 14}$ has factors of ${\displaystyle 2}$ and ${\displaystyle 7}$.

The LCM of ${\displaystyle 3,7,14}$ 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.

${\displaystyle x}$ occurs ${\displaystyle 1}$ time.

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