Let $X$ be a simply connected space with $H_2(X; \mathbb{Z}) \cong \mathbb{Z}/5$, $H_3(X; \mathbb{Z}) \cong \mathbb{Z}$ and all higher homology groups zero. Show there exists a CW-complex $Z$ with one $0$-cell, one $2$-cell and two $3$-cells and a weak homotopy equivalence $Z \to X$. (Hint: use the relative Hurewicz theorem.)
Giving the weak homotopy equivalence explicitely is probably not going to work since we know little about $X$. By cellular approximation, there is a CW-complex $Y$ and a weak homotopy equivalence $Y \to X$. By Hurewicz we have that $\pi_0(Y) \cong\pi_1(X) \cong H_0(X; \mathbb{Z}) \cong 0$, $\pi_1(Y) \cong 0$, $\pi_2(Y) \cong \mathbb{Z}/5$. So $Y$ has one connected component, and all loops are contractible. I don't see what else to do.
