2. Let $p$ be a positive odd number. Then the remainder when $p^{2}$ is divided by 8 is ( ).
(A) 1
(B) 3
(C) 5
(D) 7
2.A.

Since $p$ is a positive odd number, let $p=2k-1\left(k \in \mathbf{N}_{+}\right)$, then $p^{2}=(2k-1)^{2}=4k^{2}-4k+1=4(k-1)k+1$.
Since $k-1$ and $k$ must be one odd and one even, $(k-1)k$ must be even. Therefore, the remainder when $p^{2}$ is divided by 8 is 1.