Stage: 5 Short Challenge Level:
Given the sequence $1^5-1,\ 2^5-2,\ 3^5-3,\cdots \ n^5-n$ we see that
\[n^5 - n = n(n^4 - 1) = n(n - 1)(n + 1)(n^2 + 1)\]
and it is quite easy to see that $n(n-1)(n+1)(n^2+1)$ is divisible by $2$, $3$ and $5$ for all values of $n$. As $n$, $(n-1)$ and $(n+1)$ are three consecutive integers their product must be divisible by $2$ and by $3$. If none of these numbers is divisible by $5$ then $n$ is either of the form $5k+2$ or $5k+3$ for some integer $k$ and in both of these cases we can check that $n^2 + 1$ is
divisible by $5$. Since $2$, $3$ and $5$ are coprime therefore $n^5 - n$ is divisible by $2 \times 3 \times 5$ i.e by $30$.
Since the second term of the sequence is $2^5-2 = 30$, the divisor cannot be greater than $30$. Therefore $30$ is the largest number that divides each member of the sequence.