Solution:

Since $71$ is prime, in $cde = 71$ one of the three factors is $71$ and the other two equal $1.$ But $c$ divides $abc = 70$ and $e$ divides $efg = 72,$ and $71$ divides neither $70$ nor $72.$ So $c = e = 1$ and $d = 71.$

The system reduces to $ab = 70$ and $fg = 72.$ Each divisor $a$ of $70$ determines $b,$ giving $\tau(70) = 8$ ordered pairs, and likewise $\tau(72) = 12$ ordered pairs $(f, g).$ The total is $8 \cdot 12 = 96.$