Question

when finding the factors for 841 what is the largest factor you would have to test?

Answer 1

The square root of 841. Why? Because if there's an X that's a factor of 841, then (841/X) must also be a factor - so if you know one then you know the other, which means that once you've investigated every possibility up to and including sqrt(841), i.e. 29, then you've also investigated all the viable possibilities above it.