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Is \(x^2 y^4\) an integer divisible by 9 ?
(1) \(x\) is an integer divisible by 3
(2) \(xy\) is an integer divisible by 9
(1) \(x\) is an integer divisible by 3
(2) \(xy\) is an integer divisible by 9
16 Sep 2014, 00:58
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Official Solution:
Is \(x^2 y^4\) an integer divisible by 9 ?
The crucial observation to solve this question is to note that we are not given that \(x\) and \(y\) are integers.
(1) \(x\) is an integer divisible by 3
If \(x = 3\) and \(y=any \ integer\), then the answer would be YES. However, if \(x=3\) and \(y=\frac{1}{2}\), then the answer would be NO. Not sufficient.
(2) \(xy\) is an integer divisible by 9
If \(x = 9\) and \(y=any \ integer\), then the answer would be YES. However, if \(x=6\) and \(y=\frac{3}{2}\), then the answer would be NO. Not sufficient.
(1)+(2) We can use the same example as in the second statement:
If \(x = 9\) and \(y=any \ integer\), then the answer would be YES. However, if \(x=6\) and \(y=\frac{3}{2}\), then the answer would be NO. Not sufficient. Essentially even when taken together the statements are not sufficient to deduce that \(y\) is an integer, which would guarantee that \(x^2 y^4\) is divisible by 9. Nor can we deduce that \(y\) is such a noninteger, for which \(x^2 y^4\) is divisible by 9 (for example, \(x = 3^3\) and \(y=\frac{1}{3}\)).
Answer: E
Is \(x^2 y^4\) an integer divisible by 9 ?
The crucial observation to solve this question is to note that we are not given that \(x\) and \(y\) are integers.
(1) \(x\) is an integer divisible by 3
If \(x = 3\) and \(y=any \ integer\), then the answer would be YES. However, if \(x=3\) and \(y=\frac{1}{2}\), then the answer would be NO. Not sufficient.
(2) \(xy\) is an integer divisible by 9
If \(x = 9\) and \(y=any \ integer\), then the answer would be YES. However, if \(x=6\) and \(y=\frac{3}{2}\), then the answer would be NO. Not sufficient.
(1)+(2) We can use the same example as in the second statement:
If \(x = 9\) and \(y=any \ integer\), then the answer would be YES. However, if \(x=6\) and \(y=\frac{3}{2}\), then the answer would be NO. Not sufficient. Essentially even when taken together the statements are not sufficient to deduce that \(y\) is an integer, which would guarantee that \(x^2 y^4\) is divisible by 9. Nor can we deduce that \(y\) is such a noninteger, for which \(x^2 y^4\) is divisible by 9 (for example, \(x = 3^3\) and \(y=\frac{1}{3}\)).
Answer: E
