If w, y and z are positive integers, and w = y - z, is w a perfect square?
I. y+z is a perfect square
II. z is even
I chose E.
Dear experts please assist with explanation in this.
Thank you
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W, Y and Z; Integer Properties
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dustystormy
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People always try putting value to validate the the given conditions but somehow I like the algebraic way.
w=y-z given
St 1) y+z is a perfect square ----> assume it as k^2
therefore y = k^2 - z and substitute this condition in w
w = k^2 -2z
here w may or may not be a perfect square depending on the value of k & z
w is perfect square for k=5 & z=8 -----> w = 3
and for may other values of k & z, w will not be a perfect square
hence Insufficient
St 2) doesn't help in anyways.
hence Insufficient
Combining (1) & (2) same problem as there was in (1). hence Insufficient
therefore ANS is E
w=y-z given
St 1) y+z is a perfect square ----> assume it as k^2
therefore y = k^2 - z and substitute this condition in w
w = k^2 -2z
here w may or may not be a perfect square depending on the value of k & z
w is perfect square for k=5 & z=8 -----> w = 3
and for may other values of k & z, w will not be a perfect square
hence Insufficient
St 2) doesn't help in anyways.
hence Insufficient
Combining (1) & (2) same problem as there was in (1). hence Insufficient
therefore ANS is E
dustystormy wrote:People always try putting value to validate the the given conditions but somehow I like the algebraic way.
w=y-z given
St 1) y+z is a perfect square ----> assume it as k^2
therefore y = k^2 - z and substitute this condition in w
w = k^2 -2z
here w may or may not be a perfect square depending on the value of k & z
w is perfect square for k=5 & z=8 -----> w = 3
and for may other values of k & z, w will not be a perfect square
hence Insufficient
St 2) doesn't help in anyways.
hence Insufficient
Combining (1) & (2) same problem as there was in (1). hence Insufficient
therefore ANS is E
Hi, thanks for your response. I do not understand how you went about statement I. Especially the part where you substituted y=k^2-z into w. For me, I just plugged in numbers. But I appreciate quicker solutions.
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Target question: Is w a perfect square?Halimah_O wrote:If w, y and z are positive integers, and w = y - z, is w a perfect square?
1) y+z is a perfect square
2) z is even
IMPORTANT: Including the variable w in this question makes it look trickier than it is.
Notice that we could just as easily ask, "Is y-z a perfect square?" and IGNORE the w altogether. So, let's do that by REPHRASING our target question...
REPHRASED target question: Is y-z a perfect square?
Given: y and z are positive integers
Statement 1: y+z is a perfect square
This statement doesn't FEEL sufficient, so I'll TEST some values.
There are several values of z and z that satisfy statement 1. Here are two:
Case a: y = 5 and z = 4. Here, 5 + 4 = 9, and 9 is a perfect square. In this case y-z = 5-4 = 1, and 1 IS a perfect square
Case b: y = 7 and z = 2. Here, 7 + 2 = 9, and 9 is a perfect square. In this case y-z = 7-2 = 5, and 5 is NOT a perfect square
Since we cannot answer the REPHRASED target question with certainty, statement 1 is NOT SUFFICIENT
Aside: For more on this idea of plugging in values when a statement doesn't feel sufficient, you can read my article: https://www.gmatprepnow.com/articles/dat ... lug-values
Statement 2: z is even
No information about y.
Since we cannot answer the REPHRASED target question with certainty, statement 2 is NOT SUFFICIENT
Statements 1 and 2 combined
There are still several values of y and z that satisfy BOTH statements. Here are two:
Case a: y = 5 and z = 4. Here, 5 + 4 = 9, and 9 is a perfect square. In this case y-z = 5-4 = 1, and 1 IS a perfect square
Case b: y = 7 and z = 2. Here, 7 + 2 = 9, and 9 is a perfect square. In this case y-z = 7-2 = 5, and 5 is NOT a perfect square
Since we cannot answer the REPHRASED target question with certainty, the combined statements are NOT SUFFICIENT
Answer = E
RELATED VIDEO
REPHRASED target question: https://www.gmatprepnow.com/module/gmat ... video/1100
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Matt@VeritasPrep
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S1
y + z = x²
We want to know about y - z, so we can subtract 2z from both sides to find
y - z = x² - 2z
or
w = x² - 2z
This COULD be a perfect square, but we need to know more about z; NOT SUFFICIENT
S2:
Useless by itself!
S1 + S2:
Now we have
w = x² - 2*even
This isn't helpful either: subtracting 2*even might give us a square, but it also might not. (For example, we could have x = 4 and z = 6, or we could have x = 4 and z = 4.)
y + z = x²
We want to know about y - z, so we can subtract 2z from both sides to find
y - z = x² - 2z
or
w = x² - 2z
This COULD be a perfect square, but we need to know more about z; NOT SUFFICIENT
S2:
Useless by itself!
S1 + S2:
Now we have
w = x² - 2*even
This isn't helpful either: subtracting 2*even might give us a square, but it also might not. (For example, we could have x = 4 and z = 6, or we could have x = 4 and z = 4.)
