Zach chooses five numbers from the set {1, 2, 3, 4, 5, 6, 7} and tells their product to Claudia. She finds that this is not enough information to tell whether the sum of Zach’s numbers is even or odd. What is the product that Zach tells Claudia?

I believe it is a d1 puzzle.

Are you on my side?

The process described by Jer is exactly what I have experienced.

I believe it took me less than 20 seconds:

Anyone used to fiddle with numbers( including myself) knows that 2*6 and 4*3 are the only sample within the nine non-zero digits of of two distinct couples (a,b) & (c,d) that produce   the same result when multiplied, and in the same time  the sums have opposite parities.

Thus it boils down to mentally evaluate 1*5*7*12=60*7=420.

I estimate that I've spent over 5 min. trying to justify myself by typing the above..

But, to quote Jer, it was a TOURNAMENT (sic!) question!