I am having difficulty in solving following types of problem:
Sometimes we are given a number in terms of $n$ and we have to check whether it is divisible by a particular composite number. For example, I am posting a question here
suppose $k= n^5- n$ then prove that $k$ is divisible by $30$.
And this was my approach:
$$n^5-n= n (n^4-1) =n(n^2+1) (n+1) (n-1)$$ Since $k$ is a product of $n^2+1$ and three consecutive integers, it must be a multiple of $2$ and $3$. So it gives $k= 6m (n^2+1)$. But how can I prove that it's also a multiple of 5? And this is where I get confused.
Now, suggest some alternate way to prove above problem or some corrections in my method.