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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.

  • 2
    This turns out to be identical to [How to prove $n^5−n$ is divisible by 30 without reduction](http://math.stackexchange.com/questions/132210/how-to-prove-n5-n-is-divisible-by-30-without-reduction)2012-07-30

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