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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
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13 Sep 2010, 14:33
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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
Hope it's clear.
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
Hope it's clear.
13 Sep 2010, 14:58
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seekmba
One more thing about this question:
On GMAT when we are told that \(a\) is divisible by \(b\) (or which is the same: "\(a\) is multiple of \(b\)", or "\(b\) is a factor of \(a\)"), we can say that:
1. \(a\) is an integer;
2. \(b\) is an integer;
3. \(\frac{a}{b}=integer\).
So the terms "divisible", "multiple", "factor" ("divisor") are used only about integers (at least on GMAT).
Which means that we could omit words in red in the question. For example "is \(x^2 y^4\) an integer divisible by 9" is the same as "is \(x^2 y^4\) divisible by 9" as well as "xy is an integer divisible by 9" is the same as "xy is divisible by 9".
Hope it helps.
