Food for thought

$n^5 - n$ Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Suggestion

Find the largest integer that divides every term of the sequence $1^5-1$, $2^5-2$, $3^5-3$, …, $n^5 - n$, ….

We could try a warm-up problem. What is the largest integer that divides every term of the sequence $1^3 - 1$, $2^3 - 2$, $3^3 - 3$, …, $n^3 - n$?
We can factorise $n^5 - n = n(n^4 - 1)$. Can we factorise further by factorising $n^4 - 1$?
If $n$ is $3$ more than a multiple of $5$, then what is the remainder when we divide $n^5 - 5$ by $5$? Maybe we can generalise that.
We could make a conjecture about what the answer is; that would be of the form “Every number of the form $n^5 - n$ is divisible by $a$” (with some suitable value of $a$). Then we might try proof by induction.