1. how many correct experiments do we need to disprove a theory? how many do we need to prove a theory? explain.
2. a guidebook describes the rate of climb of a mountain trail as 120 meters per kilometer. how can you express this as a number with no units?

1.  You can never prove a theory. All theories and laws and scientific knowledge in general is provisional. If your theory is inconsistent with an experiment (and you are sure the experiment is correct), your theory needs to be pitched or modified.

2. change the meters to kilometers; 

120 m/km = .12 km/km = .12 

or the kilometers to meters; 

120 m/km = 120 m/1000m = .12 

This sounds like its telling you how far you climb vertically (.12 km) for every 1 km that you walk along the slanted, winding trail. 

The unitless number ".12" is usefull because now you can think of the rate of climb in any units you want. So you rise vertcally .12 miles for every mile you walk the trail, or you rise .12 ft for every foot you walk the trail, etc.