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.