Question

Find LCM of $\left( {6!,7!,8!} \right)$.

Answer 1

Hint: LCM is the least common multiple. LCM of three numbers is divisible by each number. Here find the values of 6, 7 and 8 factorials first and then find their LCM.

Complete step-by-step answer:
We are given to find the LCM of $\left( {6!,7!,8!} \right)$
A factorial of a number is a function that multiplies the given number by every other number below (less than) it. And while multiplying, the factorial should stop at 1.
Factorial of 6 means multiplying 6 with every number below it which is
$\Rightarrow 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720$
Factorial of 7 means multiplying 7 with every number below it which is
$\Rightarrow 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040$
Factorial of 8 means multiplying 8 with every number below it which is
$\Rightarrow 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40320$
LCM of 720, 5040, 40320 is
$\underline {720|720,5040,40320} $
     $\underline {7|} \underline {1,7,56} $
        $|1,1,8$
 LCM of 720, 5040, 40320 is
$\Rightarrow 720 \times 7 \times 8 = 40,320$
So, the correct answer is “40,320”.

Additional information.: factorial of 1 is 1 and the factorial of 0 is also 1. Factorial of a number n can also be written as $n\left( {n - 1} \right)!$.

Note: Do not confuse LCM with HCF (Highest common factor), as HCF is a factor of the given numbers and LCM is a multiple of the given numbers. It is more like, LCM is a dividend and HCF is a divisor.
Another approach :
Factorial of 7 can also be written as $7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$, where $6 \times 5 \times 4 \times 3 \times 2 \times 1$ is the value of factorial of 6 ($6!$). Therefore, factorial of 7 is $7 \times 6!$
In the same way factorial of 8 can also be written as $8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$, where $7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$ is the value of 7 factorial sand 7 factorial is $7 \times 6!$
Therefore, factorial of 8 is $8 \times 7 \times 6!$
LCM of $\left( {6!,7!,8!} \right)$ is
$\underline {|6!,7!,8!} $