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If © denotes one of three arithmetic operations, addition, subtraction

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If © denotes one of three arithmetic operations, addition, subtraction [#permalink] If © denotes one of three arithmetic operations, addition, subtraction   15 May 2024, 01:30
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­If \(©\) denotes one of three arithmetic operations, addition, subtraction, or division, and \(9©x = x\), where \(x\) is a positive integer, what is the value of \(x\)?

A. 0
B. 1
C. 2
D. 3
E. 4­

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Re: If © denotes one of three arithmetic operations, addition, subtraction [#permalink] If © denotes one of three arithmetic operations, addition, subtraction   02 Jun 2024, 02:00
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Official Solution:

If \(©\) denotes one of three arithmetic operations, addition, subtraction, or division, and \(9©x = x\), where \(x\) is a positive integer, what is the value of \(x\)?

A. 0
B. 1
C. 2
D. 3
E. 4


If \(©\) represents addition, we'd have \(9 + x = x\), which cannot be true for any value of \(x\). Hence, \(©\) is not addition.

If \(©\) represents subtraction, we'd have \(9 - x = x\), which simplifies to \(x = \frac{9}{2}\). However, since \(x\) must be a positive integer, \(©\) cannot be subtraction.

Therefore, \(©\) must represent division. In this case, \(\frac{9}{x} = x\), leading to \(x^2 = 9\). Since \(x\) is a positive integer, \(x\) must be 3.


Answer: D­
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Re: If © denotes one of three arithmetic operations, addition, subtraction [#permalink] If © denotes one of three arithmetic operations, addition, subtraction   15 May 2024, 01:56
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Bunuel wrote:
­If \(©\) denotes one of three arithmetic operations, addition, subtraction, or division, and \(9©x = x\), where \(x\) is a positive integer, what is the value of \(x\)?

A. 0
B. 1
C. 2
D. 3
E. 4­

­
9 + x = x
9 * x = x
9 / x = x
9 - x = x

Out of the above operations, only \(\frac{9 }{ x} = x\) is valid.

\(x^2 = 9\)

\(x = 3\)

Option D
 
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Re: If © denotes one of three arithmetic operations, addition, subtraction [#permalink] If © denotes one of three arithmetic operations, addition, subtraction   15 May 2024, 08:59
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If \(©\) denotes one of three arithmetic operations, addition, subtraction, or division, and \(9©x = x\), where \(x\) is a positive integer, what is the value of \(x\)?

A. 0
B. 1
C. 2
D. 3
E. 4­

For any positive integer addition and subtraction would never give the same integer over which either of these arithmetic operations are implemented.
Thus, neither addition nor subtraction is applicable in \(9©x = x\)
Only division is left.

Therefore,
\(\frac{9}{x} = x\) i.e. x^2 = 9 which gives x -3 and 3 but since x is positive, x = 3.

OR 

\(\frac{9}{x} = x = 9/3 = 3\)
­

Answer D.
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Re: If © denotes one of three arithmetic operations, addition, subtraction [#permalink]
15 May 2024, 08:59
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Bunuel
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