I would like to prove (self study) that $H^{np}(K(\mathbb{Z},n),\mathbb{Z})\neq 0$, where $n$ is even , $p\geq1$ and $K(\mathbb{Z},n)$ is the Eilenberg–Maclane space.
I used the adjunction between $[ *,K(G,n)]$ and $H^{n}(*,G)$ and I found that it suffices to prove that $[K(\mathbb{Z},n),K(\mathbb{Z},np)]\neq 0$ but I can't continue the proof. Any help is appreciated.
Let me write $B^n A$ for $K(A, n)$. The Yoneda lemma implies that $[B^n \mathbb{Z}, B^{np} \mathbb{Z}]$ is precisely the set of natural transformations $H^n(-, \mathbb{Z}) \to H^{np}(-, \mathbb{Z})$. An obvious candidate for such a natural transformation is the $p^{th}$ cup power
$$H^n(-, \mathbb{Z}) \ni \alpha \mapsto \alpha^p \in H^{np}(-, \mathbb{Z})$$
and to show that this is nonzero it suffices to find a single space on which it is nonzero, for which you can pick $\mathbb{CP}^{\infty}$.