I would like to prove that the integral cohomology of $BPU_{n}$ the classifying space of the projective unitary group of order $n$ has $n-$primary torsion.

We have a fiber sequence of the form $BSU_{n}\rightarrow BPU_{n} \rightarrow K(\mathbb{Z}/n,2)$. Then we can consider the Serre spectral sequence associated to this fibration. Let's suppose that the smallest prime dividing $n$ is a very large prime $p$. For general reasons, we know that the integral cohomology of $H^{i}(K(\mathbb{Z}/n,2))$ is $\mathbb{Z}$, 0, 0, $\mathbb{Z}/n$, 0, $\mathbb{Z}/n$, 0, $\mathbb{Z}/n$, ... where it repeats this way in a range $2p+\delta$. We also know that $H^{*}(BSU_{n})\cong \mathbb{Z}[c_{2},\dots,c_{n}]$ where deg$(c_{i})=2i$. Therefore, third page of the spectral sequences will have all its columns (but the zeroth column) conformed by zeros and cyclic groups of order $n$:

How can I answer my question using this setup?

Thanks.