3. The sequences $\left\{a_{n}\right\}$ and $\left\{b_{n}\right\}$ are both arithmetic sequences of positive integers, with common differences $d_{1}$ and $d_{2}$, respectively, and both are greater than zero. If the sets $A=\left\{a_{n} \mid n \in \mathbf{N}_{+}\right\}$ and $B=\left\{b_{n} \mid n \in \mathbf{N}_{+}\right\}$, then the first terms of the two sequences being equal (i.e., $a_{1}=b_{1}=a$) is a ( ) for the elements of the set $C=A \cap B$ to also form a positive integer arithmetic sequence.
(A) necessary but not sufficient condition
(B) sufficient but not necessary condition
(C) necessary and sufficient condition
(D) neither a sufficient nor a necessary condition
3.B.

Let $\left[d_{1}, d_{2}\right]$ denote the least common multiple of $d_{1}$ and $d_{2}$.
If $a_{1}=b_{1}=a$, then for any natural number $n$, we have $a+n\left[d_{1}, d_{2}\right] \in C$.
Conversely, for any $c_{n} \in C$, then
$$
c_{n}=a+\left(n_{1}-1\right) d_{1}=a+\left(n_{2}-1\right) d_{2} \text {, }
$$

which means $d_{2}\left|\left(n_{1}-1\right) d_{1}, d_{1}\right|\left(n_{2}-1\right) d_{2}$.
Thus, $c_{n}=a+n\left[d_{1}, d_{2}\right]$ (where $n$ is a natural number).
Therefore, the condition is sufficient.
However, the condition is not necessary. A counterexample is as follows:
$$
\begin{array}{l}
A=\{1,2, \cdots, n, \cdots\}, \\
B=\{2,3, \cdots, n+1 ; \cdots\}, \\
C=A \cap B=B, \text { but } a_{1} \neq b_{1} .
\end{array}
$$