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 $1^5-1$, $2^5-2$, $3^5-3$, …, $n^5 - n$, ….

Can you generalise your findings?

There are many ways to tackle this problem, so you might like to try more than one. Here are a few thoughts that might be helpful.

We could find the first few terms of the sequence, and look to see which numbers divide all of them.

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.