at.algebraic topology - The (n+1)-st cohomology of K(Z/p,n).

I was looking through my notes for a homotopy theory course and found the following mysterious statement (K is of course the Eilenberg-Maclane space):

$$H^{n+1}(K(mathbb Z_p,n);mathbb Z_p) cong mathbb Z_p.$$

(This would be obvious if n+1 were replaced with n. This is supposed to imply that the natural transformations $H^n(X; mathbb Z_p)to H^{n+1}(X; mathbb Z_p)$ are all multiples of the Bockstein homomorphism).

I'm at a loss trying to understand why. Spectral sequences haven't been covered yet, so there should be some simple reason. Also, is there a way to see the Bockstein in all this?

Thank you!