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05 Nov 2012, 18:18
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Difficulty:
15%
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Question Stats:
80% (01:27) correct
20%
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wrong
based on 1311
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If \(x\) and \(y\) are positive integers, and \(xy\) is divisible by 4, which of the following must be true?
A. If \(x\) is even, then \(y\) is odd.
B. If \(x\) is odd, then \(y\) is a multiple of 4.
C. If \(x+y\) is odd, then \(\frac{y}{x}\) is not an integer.
D. If \(x+y\) is even, then \(\frac{x}{y}\) is an integer.
E. \(x^y\) is even.
M08-26
A. If \(x\) is even, then \(y\) is odd.
B. If \(x\) is odd, then \(y\) is a multiple of 4.
C. If \(x+y\) is odd, then \(\frac{y}{x}\) is not an integer.
D. If \(x+y\) is even, then \(\frac{x}{y}\) is an integer.
E. \(x^y\) is even.
M08-26
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18 Oct 2023, 01:09
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If \(x\) and \(y\) are positive integers, and \(xy\) is divisible by 4, which of the following must be true?
A. If \(x\) is even, then \(y\) is odd.
B. If \(x\) is odd, then \(y\) is a multiple of 4.
C. If \(x+y\) is odd, then \(\frac{y}{x}\) is not an integer.
D. If \(x+y\) is even, then \(\frac{x}{y}\) is an integer.
E. \(x^y\) is even.
Note that the question asks which of the following MUST be true, not which COULD be true.
A. If \(x\) is even, then \(y\) is odd. This is not necessarily true. For instance, take \(x = y = 2\), which are both even.
B. If \(x\) is odd, then \(y\) is a multiple of 4. This is always true because if \(x\) is odd, then for \(xy\) to be divisible by 4, \(y\) must be a multiple of 4.
C. If \(x+y\) is odd, then \(\frac{y}{x}\) is not an integer. This is not necessarily true. For example, take \(x = 1\) and \(y = 4\).
D. If \(x+y\) is even, then \(\frac{x}{y}\) is an integer. This is not necessarily true. Consider \(x = 2\) and \(y = 4\).
E. \(x^y\) is even. This is not necessarily true. For example, consider \(x = 1\) and \(y = 4\).
Answer: B
A. If \(x\) is even, then \(y\) is odd.
B. If \(x\) is odd, then \(y\) is a multiple of 4.
C. If \(x+y\) is odd, then \(\frac{y}{x}\) is not an integer.
D. If \(x+y\) is even, then \(\frac{x}{y}\) is an integer.
E. \(x^y\) is even.
Note that the question asks which of the following MUST be true, not which COULD be true.
A. If \(x\) is even, then \(y\) is odd. This is not necessarily true. For instance, take \(x = y = 2\), which are both even.
B. If \(x\) is odd, then \(y\) is a multiple of 4. This is always true because if \(x\) is odd, then for \(xy\) to be divisible by 4, \(y\) must be a multiple of 4.
C. If \(x+y\) is odd, then \(\frac{y}{x}\) is not an integer. This is not necessarily true. For example, take \(x = 1\) and \(y = 4\).
D. If \(x+y\) is even, then \(\frac{x}{y}\) is an integer. This is not necessarily true. Consider \(x = 2\) and \(y = 4\).
E. \(x^y\) is even. This is not necessarily true. For example, consider \(x = 1\) and \(y = 4\).
Answer: B
General Discussion
05 Nov 2012, 19:14
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gmatchase
I'm happy to help with this.
First of all, this is a very challenging GMAT problem --- it really demands a great deal of number sense to answer this efficiently.
The prompt tells us x & y are positive integer (mercifully!) and x*y is divisible by 4. That could happen if either one of them is a multiple of four, or if both of them are even.
(A) Counter-example: If x = 2, then y could not be odd, because 2*(odd) will never be divisible by four. NOT ALWAYS TRUE.
(B) If x is odd, then there are no factors of 2 in x. There are two factors of 2 in x*y, since x*y is be divisible by 4. If those two factors of 2 don't come from x, they must come from y. Therefore y MUST BE divisible by four.
(C) Counter-example: x = 3 and y = 12 --- x + 12 = 15 is odd, but y/x = 12/3 = 4 is an integer. NOT ALWAYS TRUE.
(D) Counter-example: x = 12 and y = 8 ---- x + y = 20 is even, but x/y = 12/8 = 3/2 is not an integer. NOT ALWAYS TRUE.
(E) Counter-example: x = 3 and y = 4 ---- x^y = 3^4 = 81 is not even. NOT ALWAYS TRUE.
In every pair in the counter-examples, the numbers were chosen so that x*y is divisible by 4, and any additional condition in that particular choice is met.
This blog is tangentially related -- it may help a bit with number sense:
https://magoosh.com/gmat/2012/gmat-math- ... variables/
Does all this make sense?
Mike
