Question: w, x, y, and z are integers. If w > x > y > z > 0, is y a common divisor of w and x?

(1) \(\frac{w}{x}=z^{-1}+x^{-1}\)
(2)\( w^2-wy-2w=0\)

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient.
5. Statements (1) and (2) TOGETHER are not sufficient.

Correct Answer: (D)

Approach Solution : 1

Statement - 1 : w/x = (z^-1) + (x^-1)

w/x = (1/z) + (1/x)

To get wz = x+z, multiply both sides by xz.

Z can be rearranged and factored out to yield z(w1)=x. Given that w>x>z, then z=1 (if z>1, then x>w is the opposite of the stated condition).
To get w = x+1, insert z = 1 into z(w1)=x and rearrange.
W and x are two consecutive integers if w = x+1. Now, if two successive integers are co-prime, they only have one factor in common. (For instance, 20 and 21 are consecutive integers, so the only thing they have in common is 1). Because y>1 (from y>z>0) and w and x do not share any common factors greater than 1, y is not a common factor of w and x.

Therefore this statement is sufficient

Statement - 2 : (w^2) - wy -2w = 0

From this we can write, w(w−y−2)=0,

since w>0,
=> w−y−2=0
=> w−2=y

w>x>w−2 (Let us substitute y in the given inequality)

Therefore x = w−1.

Therefore this statement is sufficient.

Approach Solution : 2

w > x > y > z > 0 is known. Thus, z's minimum value is 1. y has a minimum value of 2.

Let us regard case 1 as having the following values, z = 1, y = 2, x = 3, and w = 4. Y is not a factor of both x and w in this situation.

And the values z = 1, y = 2, x = 4, and w = 8 in case 2. Y is a factor of both x and w in this situation.

Statement - 1 : w/x = (z^-1) + (x^-1)

Change the values in the equation (w/x) = (1/z) + (1/x) to reflect each case.

Case 1: The numbers make the equation work.
Case 2: The values are insufficient to solve the equation.

Therefore this statement is sufficient

Statement - 2 : (w^2) - wy -2w = 0

We can write this as w(w - y - 2) = 0
=> It is either w = 0 or (w - y - 2) = 0

w will not be 0 because w > 0. Thus, w - y = 2.

Change the values in the equation to reflect each case.

Case 1: The numbers make the equation work.
Case 2: The values are insufficient to solve the equation.

Therefore this statement is sufficient

Approach Solution : 3

Statement - 1 : w/x = (z^-1) + (x^-1)

We can write, w = (x/z) + 1
=> w−1 = x/z

We can arrange things like this on the number line: —-x ——- x/z---- w

Here x/z and w are two consecutive integers

Because there cannot be an integer(x) between two consecutive integers, we cannot have —-x/z—-x—-w as.

As a result, x/z>x => z<1.

This, however, is impossible because z must be at least 1. The only solution is therefore when x/z and x coincide. That is x/z = x => z=1.

Y (which is not equal to 1) can never be a divisor for both x and w because w and x are consecutive integers.

Therefore this statement is sufficient

Statement - 2 : (w^2) - wy -2w = 0

We know that w≠0 and w = y+2. Thus, on the number line, --y ---(y+1)----w.

Since y<x<w = x = y+1, there is only one integer between y and w. Y cannot be a divisor for both w and x, as was stated above.

Therefore this statement is sufficient

“w, x, y, and z are integers. If w > x > y > z > 0, is y a common” - is a topic of the GMAT Quantitative reasoning section of GMAT. GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

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