Find the GCD and LCM of the factorials of two given numbers

Question

Find $\gcd(20!, 12!)$ and $\text{lcm}(20!, 12!)$.

My answer is:

$20=2^2 \times 5$

$12=2^2 \times 3$

GCD $= 2^2 = 4$

LCM $= 2^2 \times 3 \times 5 = 60$

....

But my teacher said that this symbol ! means factorial. How can I find GCD and LCM for the factorial of these numbers?

Answer 1

The definition of factorial is $n!=n\times(n-1)\times\dots\times1$, or if you prefer

$$\begin{eqnarray} 0! &=& 1\\ n! &=&n\times(n-1)!\end{eqnarray}$$

Thus, if $p<q$, you have that $p!$ divides $q!$, so $GCD(p!,q!)=p!$, and $LCM(p!,q!)=q!$.

Answer 2

GCD of two factorails is equal to the smaller factorial. Why? Because the factorial is equal to product of all the numbers equal to and below 'n'. Say it was 20! And 12! . 20! Equals 20*19*18*17*16*15*14*13*(12!) GCD[20*19*18*17*16*15*14*13*(12!),(12!)]=12! LCM would be: Let the lower number be x, X!*x.