1. Let $p, q$ be natural numbers, condition A: $p^{3}-q^{3}$ is even; condition B: $p+q$ is even, then
A. A is a sufficient but not necessary condition for B
B. A is a necessary but not sufficient condition for B
C. A is a necessary and sufficient condition for B
D. A is neither a sufficient condition nor a necessary condition for B
Since $p^{3}-q^{3}=(p-q)\left(p^{2}+p q+q^{2}\right), p-q$ has the same parity as $p+q=(p-q)+2 q$, i.e.,乙 $\Rightarrow$ 甲; and $p^{2}+p q+q^{2}$ is even only when $p, q$ are both even, in which case the conclusion obviously holds, i.e., 甲 $\Rightarrow$ 乙. Therefore, the answer is $C$.