If rational and mod $p$ cohomology vanish above a certain degree, does integral cohomology also vanish? Let $X$ be a topological space. If we know that for $\mathbb{F}=\mathbb{Q}$ and $\mathbb{Z}/p$, for any prime $p$, 
$$
H^*(X;\mathbb{F})=0$$
for any $*\geq n+1$, can we conclude that
$$
H^*(X;\mathbb{Z})=0$$
for any $*\geq n+1$? If cannot what can we conclude?

If you're working with spaces having f.g. cohomology, the result in the post (not the title) holds. All $\text{Ext}(H_n(X, \mathbb{Z}), \mathbb{Z}_p)$ vanish by the universal coefficient theorem, so $H_n(X, \mathbb{Z})$ contains no torsion. It follows from the universal coefficient theorem for homology that $H_{n+1}(X, \mathbb{Z})$ must vanish, and so $H^{n+1}(X, \mathbb{Z})$ must too.
