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If \(x\) and \(y\) are positive integers, is \(x\) an odd number?
(1) \(\frac{y}{x}\) is a prime number
(2) \(x*y\) is a prime number
(1) \(\frac{y}{x}\) is a prime number
(2) \(x*y\) is a prime number
30 Sep 2021, 03:23
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Official Solution:
If \(x\) and \(y\) are positive integers, is \(x\) an odd number?
(1) \(\frac{y}{x}\) is a prime number
This one is clearly insufficient. For example, consider \(y=2\) and \(x=1=odd\) or \(y=4\) and \(x=2=even\).
(2) \(x*y\) is a prime number.
A prime number has only two factors: 1 and itself. So, \(x*y=prime\) implies that \(x=1\) and \(y=prime\) or vise-versa. So, \(x\) could be odd (when \(x=1\) and \(y=prime\)), as well, as even (when \(x=2=prime=even\) and \(y=1\)) Not sufficient.
(1)+(2) The case when \(y=1\) and \(x=prime\) (from (2)) is not possible because in this case \(\frac{y}{x}\) would be a proper fraction (a number between 0 and 1), so not a prime as given in (1). Hence we have another case from (2): \(x=1=odd\) and \(y=prime\). Sufficient.
Answer: C
If \(x\) and \(y\) are positive integers, is \(x\) an odd number?
(1) \(\frac{y}{x}\) is a prime number
This one is clearly insufficient. For example, consider \(y=2\) and \(x=1=odd\) or \(y=4\) and \(x=2=even\).
(2) \(x*y\) is a prime number.
A prime number has only two factors: 1 and itself. So, \(x*y=prime\) implies that \(x=1\) and \(y=prime\) or vise-versa. So, \(x\) could be odd (when \(x=1\) and \(y=prime\)), as well, as even (when \(x=2=prime=even\) and \(y=1\)) Not sufficient.
(1)+(2) The case when \(y=1\) and \(x=prime\) (from (2)) is not possible because in this case \(\frac{y}{x}\) would be a proper fraction (a number between 0 and 1), so not a prime as given in (1). Hence we have another case from (2): \(x=1=odd\) and \(y=prime\). Sufficient.
Answer: C
