We have:

\(n^5-n=n\left(n^4-1\right)=n\left(n^2+1\right)\left(n^2-1\right)\)

1. Prove that: A divisible by 3

- If n divisible by 3 => A divisible by 3

- If n\(⋮̸\)3 => \(n^2\equiv1\left(mod3\right)\) => n² - 1 divisible by 3 => A divisible by 3

So A divisible by 3 

2. Prove that A divisible by 5

- If n divisible by 5 => A divisible by 5

- If n \(⋮̸\)5 => \(n^2\equiv1,4\left(mod5\right)\)

 + If n² \(\equiv1\left(mod5\right)\Rightarrow n^2-1⋮5\) => A divisible by 5

  + If n² \(\equiv4\left(mod5\right)\)=> \(n^2+1⋮5\) => A divisible by 5

So A divisible by 5

\(But\left(3,5\right)=1\)

=> A divisible by 3.5 = 15

Or A divisible by 15