2023/023) How do you prove that $n^4−1$ is divisible by 5 if n is not a multiple of 5?

We have $n^4 -1 = (n^2+1)(n^2–1) = (n^2–4 + 5)(n^2–1) = (n^2–4)(n^2–1) + 5(n^2–1)$ 

$= (n+2)(n-2)(n+1)(n-1) + 5(n^2–1)$

now the 2nd term is multiple of 5 and 1st term is product of 4 numbers which along with are 5 consecutive numbers. so one of them has to be divisible by 5. but as n is not divisible by 5 so one of the 4 other numbers is a multiple of 5 and hence the product. as given number is sum of 2 numbers each multiple of 5 and hence the given expression.