Consider the "$m$-th power" map $f:K(\mathbb Z,n)\to K(\mathbb Z,n)$ given by $m\in \mathbb Z\cong H^n(K(\mathbb Z,n),\mathbb Z)\cong [K(\mathbb Z,n), K(\mathbb Z,n)]$. Is it true that in any degree the map $f^*$ on integral cohomology sends any element to a multiple of $m$? It's obviously true in degree $n$ where $f^*$ is just multiplication by $m$ but what about higher degrees? It's enough to consider the case when $m=p$ is prime.