2022/010) Solve the equation $n! = n^3 - n$ for integer n

Note: I picked the question from  https://www.youtube.com/watch?v=uqaRfADGOtg and I provide my solution for the same in a diffrerent way.

 We see that $n!$ grows at a rate much faster than $n^3$ 

For solving this we can  fix the bound for n and check on the same.

as we see $6^3 = 216$ and $6! = 720$ which is greater

by chceking the values of n from 0 to 5 we get n = 5 satsfies the equation

so Ans $n= 5$