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16 Sep 2014, 00:38
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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.
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.
16 Sep 2014, 00:38
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Official Solution:
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
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
18 Oct 2023, 01:06
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I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.
