mathematics - How many digits can be removed from a multiplication puzzle and still give only one answer?

There's a common category of mathematical puzzle which involves determining missing numbers in a long multiplication problem.

As an example from this site (problem 10):

       1,7__ | (    1736)
 x     _,_43 | (    5843)
 ----------- | 
       5,20_ | (    5208)
      69,440 | (   69440)
   1,38_,800 | ( 1388800)
 + _,680,0_0 | ( 8680000)
 ----------- | 
  1_,14_,448 | (10143448)

It's possible without too much difficulty to deduce the original multiplication problem. However, it's fairly easy to see that not all of these numbers are required to achieve the same result. Trivially:

       1,7__
 x     _,_43
 -----------
       5,20_
      _9,44_
   1,38_,8__
 + _,680,___
 -----------
  1_,14_,448

gives the same result. For a given multiplication problem, how many numbers can I maximally remove to still allow only one unique solution that can determined without brute-force guessing?