Determine the cohomology ring $H^*(\mathbb{R}P^n; \mathbb{Z}).$
I was given a hint to first calculate the cohomology groups $$ H^k(\mathbb{R}P^n;\mathbb{Z}).$$ And that I may use $$ H^*(\mathbb{R}P^n;\mathbb{Z}/2) \cong (\mathbb{Z}/2)[x] /(x ^(n+1)).$$ And that my answer will depend on the parity of $n.$
I also found this link here : integral cohomology ring of real projective space
Still I am unable to fill in all details.
1- How my answer will depend on the parity of $n$?
2- How I And that I may use $$ H^*(\mathbb{R}P^n;\mathbb{Z}/2) \cong (\mathbb{Z}/2)[x] /(x ^(n+1)).$$
Could anyone help me in solving this please?