30-1 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.
(China High School Mathematics League, 1983)
[Solution] $p^{3}-q^{3}$ is even $\Longleftrightarrow p, q$ have the same parity $\Longleftrightarrow p+q$ is even. Therefore, the answer is $(C)$.