Solve for f 1/f=1/p+1/q

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

Since ${\displaystyle f,p,q}$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part ${\displaystyle 1,1,1}$ then find LCM for the variable part ${\displaystyle f^1,p^1,q^1}$.

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

The number ${\displaystyle 1}$ is not a prime number because it only has one positive factor, which is itself.

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

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

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

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

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

Multiply ${\displaystyle fp}$ by ${\displaystyle q}$.

Multiply each term in ${\displaystyle \frac{1}{f}=\frac{1}{p}+\frac{1}{q}}$ by ${\displaystyle fpq}$.

Simplify the left side.

Simplify the right side.

Rewrite the equation as ${\displaystyle fq+fp=pq}$.

Factor ${\displaystyle f}$ out of ${\displaystyle fq+fp}$.

Divide each term in ${\displaystyle f(q+p)=pq}$ by ${\displaystyle q+p}$ and simplify.