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Is \(\frac{x + y}{2}\) an integer?
(1) \(|x| = \sqrt {y^2}\)
(2) \(xy = 9\)
(1) \(|x| = \sqrt {y^2}\)
(2) \(xy = 9\)
17 Jul 2017, 05:56
Kudos
Bookmarks
Official Solution:
Is \(\frac{x + y}{2}\) an integer?
First of all, note that we are NOT told that \(x\) and \(y\) are integers.
(1) \(|x| = \sqrt {y^2}\)
The above implies that \(|x| = |y|\). So, \(x=-y\) or \(x=y\). If \(x=-y\), then \(\frac{x + y}{2}=\frac{-y + y}{2}=0\), which is an integer. But if \(x=y\), then \(\frac{x + y}{2}=\frac{x + x}{2}=x\). This will be an integer if \(x\) is an integer, but it won't be an integer if \(x\) is not an integer. Not sufficient.
(2) \(xy = 9\).
If \(x=y=3\), then \(\frac{x + y}{2}=\frac{3 + 3}{2}=3\), which is an integer. But if \(x=\frac{1}{3}\) and \(y=27\), then \(\frac{x + y}{2}\) is not an integer. Not sufficient.
(1)+(2) If, from (1), \(x=-y\), then from (2) we'll have \(-x^2=9\), which is the same as \(x^2=-9\) but this is not possible (the square of a number cannot be negative). Thus, it must be true that \(x=y\), so \(x^2=9\), which gives \(x=y=3\) or \(x=y=-3\). In either case, \(\frac{x + y}{2}\) is an integer. Sufficient.
Answer: C
Is \(\frac{x + y}{2}\) an integer?
First of all, note that we are NOT told that \(x\) and \(y\) are integers.
(1) \(|x| = \sqrt {y^2}\)
The above implies that \(|x| = |y|\). So, \(x=-y\) or \(x=y\). If \(x=-y\), then \(\frac{x + y}{2}=\frac{-y + y}{2}=0\), which is an integer. But if \(x=y\), then \(\frac{x + y}{2}=\frac{x + x}{2}=x\). This will be an integer if \(x\) is an integer, but it won't be an integer if \(x\) is not an integer. Not sufficient.
(2) \(xy = 9\).
If \(x=y=3\), then \(\frac{x + y}{2}=\frac{3 + 3}{2}=3\), which is an integer. But if \(x=\frac{1}{3}\) and \(y=27\), then \(\frac{x + y}{2}\) is not an integer. Not sufficient.
(1)+(2) If, from (1), \(x=-y\), then from (2) we'll have \(-x^2=9\), which is the same as \(x^2=-9\) but this is not possible (the square of a number cannot be negative). Thus, it must be true that \(x=y\), so \(x^2=9\), which gives \(x=y=3\) or \(x=y=-3\). In either case, \(\frac{x + y}{2}\) is an integer. Sufficient.
Answer: C
