swerve 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.
The OA is E
Source: Manhattan Prep
The optimum approach would be to test values.
Given: \(w, y\), and \(z\) are positive integers, and \(w = y - z\)
We have to find out whether \(w\) is a perfect square.
Let's take each statement one by one.
1) \(y + z\) is a perfect square.
Case 1: Say y = 5 and z = 4. We see that y + z = 5 + 4 = 9, a perfect square and w = y - z = 5 -4 = 1, a perfect square. The answer is yes.
Case 2: Say y = 7 and z = 2. We see that y + z = 7 + 2 = 9, a perfect square but w = y - z = 7 - 2 = 5, NOT a perfect square. The answer is no.
No unique answer. Insufficient.
2) \(z\) is even.
No information about w and y. Both the cases discussed in Statement 1 are applicable here, too. Insufficient.
(1) and (2) together
Both the cases discussed in Statement 1 are applicable here, too. Insufficient.
The correct answer:
E
Hope this helps!
-Jay
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