BALTIC-WAY 1997 Problem 8

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?