6. On each face of a cube, write a positive integer, and at each vertex, write the product of the positive integers on the three faces meeting at that vertex. If the sum of the numbers written at the eight vertices is 2014, then the sum of the numbers written on the six faces is $\qquad$ .


6. 74 .

As shown in Figure 4, let the pairs of positive integers written on the opposite faces of the cube be $(a, b),(c, d),(e, f)$.

Then the sum of the numbers written at each vertex is
$$
\begin{array}{l}
a c f+a d f+a c e+a d e+b c f+b c e+b d f+b d e \\
=(a+b)(c+d)(e+f) \\
=2014=2 \times 19 \times 53 .
\end{array}
$$

Thus, the sum of the numbers written on each face is
$$
\begin{array}{l}
(a+b)+(c+d)+(e+f) \\
=2+19+53=74 .
\end{array}
$$