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31 Dec 2023, 09:09
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If \(©\) denotes one of three arithmetic operations, addition, subtraction, or division, and if \(x\) is a positive integer, what is the value of \(x\)?
(1) \(1©x = x\)
(2) \(2©x > x©2\)
(1) \(1©x = x\)
(2) \(2©x > x©2\)
31 Dec 2023, 09:09
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Official Solution:
If \(©\) denotes one of three arithmetic operations, addition, subtraction, or division, and if \(x\) is a positive integer, what is the value of \(x\)?
(1) \(1©x = x\)
If \(©\) represents addition, we'd have \(1 + x = x\), which cannot be true for any value of \(x\). Hence, \(©\) is not addition.
If \(©\) represents subtraction, we'd have \(1 - x = x\), which simplifies to \(x = \frac{1}{2}\). However, since \(x\) must be a positive integer, \(©\) cannot be subtraction.
Therefore, \(©\) must represent division. In this case, \(\frac{1}{x} = x\), leading to \(x^2 = 1\). Since \(x\) is a positive integer, \(x\) must be 1.
Sufficient.
(2) \(2©x > x©2\)
If \(©\) represents addition, we'd have \(2 + x > x + 2\), which is not true for any value of \(x\). Hence, \(©\) is not addition.
If \(©\) represents subtraction, we'd have \(2 - x > x - 2\), simplifying to \(x < 2\). Given that \(x\) is a positive integer, if \(©\) is subtraction, then \(x\) must be 1.
If \(©\) represents division, we'd have \(\frac{2}{x} > \frac{x}{2}\). Cross-multiplying (we can safely do that since \(x\) is a positive integer) gives \(x^2 < 4\), which narrows down to \(-2 < x < 2\). As \(x\) is a positive integer, \(x\) must be 1. Therefore, if \(©\) is division, \(x\) is 1.
In both viable cases (subtraction or division), \(x\) turns out to be 1.
Answer: D
If \(©\) denotes one of three arithmetic operations, addition, subtraction, or division, and if \(x\) is a positive integer, what is the value of \(x\)?
(1) \(1©x = x\)
If \(©\) represents addition, we'd have \(1 + x = x\), which cannot be true for any value of \(x\). Hence, \(©\) is not addition.
If \(©\) represents subtraction, we'd have \(1 - x = x\), which simplifies to \(x = \frac{1}{2}\). However, since \(x\) must be a positive integer, \(©\) cannot be subtraction.
Therefore, \(©\) must represent division. In this case, \(\frac{1}{x} = x\), leading to \(x^2 = 1\). Since \(x\) is a positive integer, \(x\) must be 1.
Sufficient.
(2) \(2©x > x©2\)
If \(©\) represents addition, we'd have \(2 + x > x + 2\), which is not true for any value of \(x\). Hence, \(©\) is not addition.
If \(©\) represents subtraction, we'd have \(2 - x > x - 2\), simplifying to \(x < 2\). Given that \(x\) is a positive integer, if \(©\) is subtraction, then \(x\) must be 1.
If \(©\) represents division, we'd have \(\frac{2}{x} > \frac{x}{2}\). Cross-multiplying (we can safely do that since \(x\) is a positive integer) gives \(x^2 < 4\), which narrows down to \(-2 < x < 2\). As \(x\) is a positive integer, \(x\) must be 1. Therefore, if \(©\) is division, \(x\) is 1.
In both viable cases (subtraction or division), \(x\) turns out to be 1.
Answer: D
13 May 2024, 07:58
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