Question 16
If we add $1996$ and $1997$, we first add the unit digits $6$ and $7$. Obtaining $13$, we write down $3$ and "carry" $1$ to the next column. Thus we make a carry. Continuing, we see that we are to make three carries in total:
$$
\begin{array}{r}
111 \\
1996 \\
+1997 \\\hline
3993
\end{array}
$$
Does there exist a positive integer $k$ such that adding $1996 \cdot k$ to $1997 \cdot k$ no carry arises during the whole calculation?
$$
\begin{array}{r}
111 \\
1996 \\
+1997 \\\hline
3993
\end{array}
$$
Does there exist a positive integer $k$ such that adding $1996 \cdot k$ to $1997 \cdot k$ no carry arises during the whole calculation?