[email protected] wrote:w, x, y, and z are integers. If w > x > y > z > 0, is y a common divisor of w and x?
(1) w/x = z^-1 + x^-1
(2) w² - wy - 2w = 0
TRICKY question! I couldn't think of a super nice approach, so please forgive the length of this solution.
Important Concept:
If integer k is greater than 1, and if k is a factor (divisor) of N, then k is not a divisor of N+1
For example, since 7 is a factor of 350, we know that 7 is not a factor of (350
+1)
Similarly, since 8 is a factor of 312, we know that 8 is not a factor of 313
We can extend this concept to say:
If integer k is greater than 1, then k cannot be a divisor of both N and N+1
In other words,
if integer k is greater than 1, then k cannot be a common divisor of two consecutive integers.
Okay, now onto the question . . .
Target question: Is y a common divisor of w and x?
Given: w > x > y > z > 0
IMPORTANT: From this, we can conclude that
y > 1, which means we may be able to apply the
green rule above.
Statement 1: w/x = z^-1 + x^-1
So, w/x = 1/z + 1/x
Rearrange terms to get: w/x - 1/x = 1/z
Rewrite to get: (w - 1)/x = 1/z
Cross multiply to get: z(w - 1) = x
IMPORTANT: This equation tells us that z = 1. Here's why.
Since it's given that x < w, we can replace x with z(w - 1) to get: z(w - 1) < w
Now divide both sides by (w - 1) to get: z < w/(w-1)
w/(w-1) is just slightly bigger than 1, and since we're told that z is positive, it MUST be the case that
z = 1.
If z = 1, and z(w - 1) = x, then we can see that w - 1 = x
Or we can write, w = x + 1
This tells us that w is 1 greater than x, which means
x and w are consecutive integers.
If y > 1, and x and w are consecutive integers, then (by the
green rule above), we can be certain that
y is definitely not a common divisor of w and x
Since we can answer the
target question with certainty, statement 1 is SUFFICIENT
Statement 2: w² - wy - 2w = 0
Factor to get: w(w - y - 2) = 0
This means that EITHER w = 0 OR w - y - 2 = 0
w cannot equal 0 since we're told that w > x > y > z > 0
So, it MUST be the case that w - y - 2 = 0
Rearrange to get w - y = 2
In other words,
w is 2 greater than y
Since we're told that w > x > y > z > 0, we know that x must lie between the values of w and y, which means
x and w are consecutive integers.
If y > 1, and x and w are consecutive integers, then (by the
green rule above), we can be certain that
y is definitely not a common divisor of w and x
Since we can answer the
target question with certainty, statement 2 is SUFFICIENT
Answer =
D
Cheers,
Brent
!!
