Hatcehr Exercise 3.2.16.
Show that if $X$ and $Y$ are finite CW complexes such that $H^∗(X;\mathbb Z)$ and $H^∗(Y;\mathbb Z)$ contain no elements of order a power of a given prime $p$, then the same is true for $X×Y$ . [Apply Theorem 3.15 with coefficients in various fields.]
The hint says we can apply
Theorem 3.15. The cross product $H^∗(X;R)⊗_R H^∗(Y;R)→H^∗(X×Y;R)$ is an isomorphism of rings if $X$ and $Y$ are CW complexes and $H^k(Y;R)$ is a finitely generated free $R$-module for all $k$.
Take an element $a\in H^∗(X×Y)$ with order $p^n$. I need to produce some elements with orders powers of $p$ in $H^∗(X;\mathbb Z)$ and $H^∗(Y;\mathbb Z)$.
How can I use his hint? I am thinking about taking $R=\mathbb Z_p$ or $\operatorname{GF}(p^n)$ but I don't see the connection to the right solution.