= Solution
The standard <CW complex> structure on <Real projective space> $\mathbb{RP}^n$ has one cell in each dimension $0,1,\ldots,n$. Its <cellular chain complex> has $C_q\cong\mathbb Z$ and differential
$$
d_q=1+(-1)^q=
\begin{cases}
2,&q\text{ even},\\
0,&q\text{ odd}.
\end{cases}
$$
Consequently
$$
H_q(\mathbb{RP}^n;\mathbb Z)\cong
\begin{cases}
\mathbb Z,&q=0,\\
\mathbb Z/2,&0<q<n\text{ and }q\text{ odd},\\
\mathbb Z,&q=n\text{ and }n\text{ odd},\\
0,&\text{otherwise}.
\end{cases}
$$
After tensoring the cellular complex with $\mathbb F_2=\mathbb Z/2$, every differential vanishes, so
$$
H_q(\mathbb{RP}^n;\mathbb F_2)\cong
\begin{cases}
\mathbb F_2,&0\leq q\leq n,\\
0,&q>n.
\end{cases}
$$
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