Victor Loves Eight

Each integer can be written as $\overline{x_1x_2x_3x_4x_5x_6x_7x_8}$ , where each $x_t = 0,1,\cdots,8$ and $x_1+\cdots+x_8 = 8$ . The number of non-negative integer solutions to the above equation is $\binom{8 + 8 - 1}{8} = \binom{15}{8} = 6435 = 3^2 \times 5 \times 11 \times 13.$ So the largest prime factor of $x$ is $\boxed{13}$ .