2023/013) How do I show that for any natural number n, the result of $1^{1987}+2^{1987}+\cdots+n^{1987} $ is not divisible by $n+ 2$

 Let us consider $2^{1987}+3^{1987}+\cdots+(n)^{1987} $

We have $k^{1987} + (n+2-k) ^{1987 }$ is divisible by n+ 2

If n is odd  taking k from 2 to $\frac{n-1}{2}$ we have |frac{n-1}{2}$ pairs and eac pair is divisible by n+2 and adding  1 does not divide by 1 

If n is even taking k from 2 to $\frac{n}{2}$ we have $\frac{n}{2}$ pairs and each pair is divisible by n+2 and middle number is $(\frac{n+2}{2})^{1987} $ 

If n+ 2 is a multiple of 4 this is even and adding 1 makes it odd and hence the sum is not divisible by n+ 2

If n+2 is of the form 4m+ 2 and in this case divisible by 2m+ 1so adding 1 does not make it divisible by 2m+ 1 so not divisible by n+ 2  proved