Solution:

Since $a$ divides $b$ and $c,$ it divides $a + b + c = 100.$ Write $b = as$ and $c = at$ with $s, t \ge 1;$ then $a(1 + s + t) = 100,$ so $$s + t = \frac{100}{a} - 1.$$ For positive $s$ and $t$ we need $\frac{100}{a} \ge 3,$ so $a \in \{1, 2, 4, 5, 10, 20, 25\}.$

For each such $a,$ the equation $s + t = \frac{100}{a} - 1$ has $\frac{100}{a} - 2$ ordered positive solutions. Summing, $$(100 + 50 + 25 + 20 + 10 + 5 + 4) - 2 \cdot 7 = 214 - 14 = 200.$$