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16 Sep 2014, 00:41
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If \(x\) and \(y\) are positive integers and \(\frac{x}{y}\) is an even integer, which of the following can be an odd integer?
A. \(x\)
B. \(xy\)
C. \(x - y\)
D. \(x + 2y\)
E. \(\frac{x}{3}\)
A. \(x\)
B. \(xy\)
C. \(x - y\)
D. \(x + 2y\)
E. \(\frac{x}{3}\)
16 Sep 2014, 00:41
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Official Solution:
If \(x\) and \(y\) are positive integers and \(\frac{x}{y}\) is an even integer, which of the following can be an odd integer?
A. \(x\)
B. \(xy\)
C. \(x - y\)
D. \(x + 2y\)
E. \(\frac{x}{3}\)
Number picking:
If \(x=2\) and \(y=1\), only option C results in an odd value: \(x-y = 2-1 = 1\), which is odd.
Process of Elimination (POE):
Given \(\frac{x}{y} = \text{even}\), it follows that \(x = y*\text{even} = \text{even}\).
A. \(x\). We've established that \(x\) is even. Discard.
B. \(xy\). Since \(x\) is even, \(xy\) will also be even. Discard.
C. \(x-y\). Keep.
D. \(x + 2y\). The expression \(x + 2y\) evaluates to \(\text{even} + \text{even}*y\), which is always even. Discard.
E. \(\frac{x}{3}\). Dividing an even number by an odd integer either yields an even integer, like \(\frac{6}{3} = 2\), or a non-integer, such as \(\frac{2}{3}\). Discard.
The only option remaining is C, hence it must be correct.
Answer: C
If \(x\) and \(y\) are positive integers and \(\frac{x}{y}\) is an even integer, which of the following can be an odd integer?
A. \(x\)
B. \(xy\)
C. \(x - y\)
D. \(x + 2y\)
E. \(\frac{x}{3}\)
Number picking:
If \(x=2\) and \(y=1\), only option C results in an odd value: \(x-y = 2-1 = 1\), which is odd.
Process of Elimination (POE):
Given \(\frac{x}{y} = \text{even}\), it follows that \(x = y*\text{even} = \text{even}\).
A. \(x\). We've established that \(x\) is even. Discard.
B. \(xy\). Since \(x\) is even, \(xy\) will also be even. Discard.
C. \(x-y\). Keep.
D. \(x + 2y\). The expression \(x + 2y\) evaluates to \(\text{even} + \text{even}*y\), which is always even. Discard.
E. \(\frac{x}{3}\). Dividing an even number by an odd integer either yields an even integer, like \(\frac{6}{3} = 2\), or a non-integer, such as \(\frac{2}{3}\). Discard.
The only option remaining is C, hence it must be correct.
Answer: C
22 Oct 2023, 02:36
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I've revised the question and solution, incorporating additional details for improved clarity. I trust this makes it more comprehensible.
