Find the LCM 7a^2 , 9ab^3 , 21abc^4

Since ${\displaystyle 7a^2,9a{b}^3,21ab{c}^4}$ contain both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part ${\displaystyle 7,9,21}$ then find LCM for the variable part ${\displaystyle a^2,a^1,b^3,a^1,b^1,c^4}$.

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

${\displaystyle 7}$ is a prime number

${\displaystyle 9}$ has factors of ${\displaystyle 3}$ and ${\displaystyle 3}$.

${\displaystyle 21}$ has factors of ${\displaystyle 3}$ and ${\displaystyle 7}$.

The LCM of ${\displaystyle 7,9,21}$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

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

${\displaystyle a}$ occurs ${\displaystyle 2}$ times.

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

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

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

${\displaystyle b}$ occurs ${\displaystyle 3}$ times.

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

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

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

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

The factors for ${\displaystyle c^4}$ are ${\displaystyle c\cdot c\cdot c\cdot c}$, which is ${\displaystyle c}$ multiplied by each other ${\displaystyle 4}$ times.

${\displaystyle c}$ occurs ${\displaystyle 4}$ times.

The LCM of ${\displaystyle a^2,a^1,b^3,a^1,b^1,c^4}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.