To evaluate the expression \((83)_n = 812\), we need to understand what this means: \( (83)_n \) represents the number \( 83 \) in base \( n \).
In base \( n \): \[ (83)_n = 8 \cdot n^1 + 3 \cdot n^0 = 8n + 3 \]
We need to set this equal to \( 812 \): \[ 8n + 3 = 812 \]
Next, we will solve for \( n \):
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Subtract \( 3 \) from both sides: \[ 8n = 812 - 3 \] \[ 8n = 809 \]
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Divide both sides by \( 8 \): \[ n = \frac{809}{8} = 101.125 \]
This value of \( n \) is not a whole number, indicating an inconsistency with the integer base representation or transcription error in interpreting the base digits.
Since \( 8 \) is not a permissible digit in any base less than \( 9 \), we can re-evaluate the possible valid values from the provided options. The only base in which \( (83)_n \) could maintain the representation is a base that could encompass both digits (i.e., \( 0-8 \)) and exceed the potential maximum base for valid decimal (which would typically be base \( 9 \)).
Checking provided responses
Calculate \( n \) for the valid options:
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n = 9: \[ (83)_9 = 8 \cdot 9 + 3 = 72 + 3 = 75 \quad \text{(not equal to 812)} \]
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n = 4: \[ (83)_4 = 8 \cdot 4 + 3 = 32 + 3 = 35 \quad \text{(not equal to 812)} \]
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n = 2: \[ (83)_2 = 8 \cdot 2 + 3 = 16 + 3 = 19 \quad \text{(not equal to 812)} \]
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n = 1: (not valid base for representation with digits > 1) \[ (n = 1) \rightarrow not applicable. \]
So the only base option that could offer a valid representation under calculated conversions through integers needs clarification as it doesn't hold equality check leading to a numeric resolution mistake or misconception.
Conclusion:
None of the given answers produce \( 812 \) when computed, indicating that there might have been misinterpretation or different facets in digits leading towards specific bases where \( (83)_n \) holds value under conversion standards beyond conceivable accounts leading to clarifying truly misaligned items with expected interpretations of final evaluations.
In context, please check constraints against valid interpretations of possible listed bases compared to segments possible through resultant percentage markings \( (as none provide result equating to 812). \)
11 answers