There can only be 2 zeros on the end.  Here's why.

The only way to get a zero on the end is to multiply by 10.
10 has prime factors 2 and 5.  With N = pq × qr × rp, there 
are two factors p, two factors q, and two factors r.  So one
pair of like factors must be 2's and another pair 5's.  
Regardless of what the third prime factor is, we can have 00 
on the end only if we have a factor of 100 = 2×2×5×5. The 
smallest number that would have two 0's on the end is

N = pq × qr × rp = (2×3) × (3×5) × (5×2) = 900.

All such integers are of the form (2×3) × (3×r) × (r×2) where 
r is any prime larger than 3.  No factor of 1000 is possible,
for that would take a product of three 2's and three 5's.

Edwin