In the Wikipedia article about the Whitehead theorem, it is said that
Combining [the Whitehead theorem for homotopy] with the Hurewicz theorem yields a useful corollary: a continuous map $f\colon X\to Y$ between simply connected CW complexes that induces an isomorphism on all integral homology groups is a homotopy equivalence.
However, I have trouble understanding how to arrive at this corollary: The Hurewicz theorem only makes a statement about $(n-1)$-connected CW complexes, i.e. complexes $X$ with $\pi_i(X)=0$ for all $i\leq n-1$.
As noted in Weak Homotopy Equivalence Induces Isomorphisms on Homology, we can replace $Y$ by the mapping cylinder of $f$; the effect is that we can assume that $f$ is an inclusion. Then we have long exact sequences for the pair $(Y, X)$ in homotopy and homology. If $f$ induces an isomorphism on homology in all dimensions, then $H_n(Y,X) =0$ for all $n$. Combined with simple connectivity and the relative Hurewicz theorem, we see that $\pi_n(Y,X)=0$ for all $n$, which means that $f$ induces an isomorphism on homotopy groups, so it is a homotopy equivalence.