Interview q: Small possible length of stick from an array of stick lengths
Pragmatic approach: If I were given this in an interview, I'd formulate it as an integer linear programming problem and let an off-the-shelf ILP solver look for a solution for me.
Theoretical approach: As Dmitry mentions, this problem is related to bin packing.  (It's not identical because you require the bins to be full, whereas normal bin packing doesn't require that.)  Wikipedia says that there is a pseudopolynomial time algorithm for bin packing when the bin size is fixed, so you might look up that algorithm and see if it can be modified to apply to your problem.  You might also explore this paper:
Michel X. Goemans, Thomas Rothvoß. &quot;Polynomiality for bin packing with a constant number of item types&quot;. SODA 2014.