siddharthmuzumdar wrote:
First of all, Ian, wonderful approach. I learned a lot from your approach. Thanks.
I followed the following crude approach to solve the problem:
We know that w > x > y > z > 0. Thus the least value of z is 1. Least value of y is 2.
I considered the values of z, y, x and w as
Case 1. z = 1, y = 2, x = 3, w = 4. In this case, y is NOT a factor of both x and w
Case 2. z = 1, y = 2, x = 4, w = 8. In this case, y is a factor of both x and w
Statement 1: Substitute the values for each case in the equation given (w/x) = (1/z) + (1/x)
Case 1: The values satisfy the equation.
Case 2: The values do not satisfy the equation.
Statement 2: w^2 - wy - 2w = 0 => w(w - y - 2) = 0 => either w = 0 or (w - y - 2) = 0.
w cannot be 0 because w > 0. Thus, w - y = 2.
Substitute the values for each case in the equation above.
Case 1: The values satisfy the equation.
Case 2: The values do not satisfy the equation.
As seen from both statements, only 1 case satisfies the given statements. Thus, the given numbers are consecutive integers. We can see that y is, thus, not a factor of both x and w.
Thus, D is the answer.
This is how I solved it. I read through the other solutions in this thread but I find solving this question backwards is easier.