OBM 1994 Problem 5

Call a super-integer an infinite sequence of decimal digits: $\ldots d_n \ldots d_2d_1$ .

(Formally speaking, it is the sequence $(d_1,d_2d_1,d_3d_2d_1,\ldots)$ )

Given two such super-integers $\ldots c_n \ldots c_2c_1$ and $\ldots d_n \ldots d_2d_1$ , their product $\ldots p_n \ldots p_2p_1$ is formed by taking $p_n \ldots p_2p_1$ to be the last n digits of the product $c_n \ldots c_2c_1$ and $d_n \ldots d_2d_1$ . Can we find two non-zero super-integers with zero product? (a zero super-integer has all its digits zero)