What kind of equation is this?

[Guest]

Consider all pairs of positive integers w and z whose sum is 5. For how many values of w does there exist a positive integer x that satisfies both 2w = x and xz = 64?
A. 0
B. 2
C. 4
D. 8
E. 12
F. Infinitely many

^^^^So. What kind of problem is this, and what level of math is it based on?
Bonus points if you can show the steps it takes to solve the problem.

[Guest]

A. Zero values of w satisfy all the constraints.

I guess technically this would be a non-linear integer programming problem (because you have a product of two variables in one of your equations, and constraints that say all values are integers). You can solve it by exhaustively trying out all values of w and z, but it is easy enough to show that if a solution existed then 2wz=64 (because x=2w), which means wz=32 (divided both sides by 2), and since w and z are between 0 and 5 the largest wz could possibly be is [tex]5\times 5=25[/tex] which is less than the target of 32.

Generally speaking integer programming problems are very hard to solve, so usually you need to do some algebraic manipulation to simplify things, and if you are lucky you will be left with something you can exhaustively search.

Hope this helped,

R. Baber.