1. Let $p, q$ be natural numbers, condition A: $p^{3}-q^{3}$ is an even number; condition B: $p+q$ is an even number, 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
(C)
1.【Analysis and Solution】If $p^{3}-q^{3}$ is an even number $\Leftrightarrow p, q$ have the same parity $\Leftrightarrow p+q$ is an even number.
【Extension】This problem tests basic knowledge of number theory congruence: $a+b \equiv a-b(\bmod 2), a \equiv a^{n}(\bmod 2)$. Using these, we can solve the following problem:

In $1^{2}, 2^{2}, \cdots, 1983^{2}$, if plus or minus signs are added arbitrarily, is the resulting algebraic sum odd or even? In fact, regardless of how the signs are added, the resulting algebraic sum $\equiv 1+2+\cdots+1983 \equiv 0(\bmod 2)$, i.e., it is always even.