Whose camp are you in?

40 students attended the brilliant summer camp at Stanford University. \(k\) of the students at the camp are working on math problems and the others are working on non-math problems. One of the camp mentors noticed that whenever all the students were partitioned into 2 or more groups of equal size, they could never have the same number of math students in every group. How many different values could \(k\) have?

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