In a certain numeral system, the number '123' represents the decimal number 27.

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Question: In a certain numeral system, the number \'123\' represents the decimal number 27. What is the value of \'321\' in the same system?

Options:

Correct Answer: 81

Solution:

\'123\' in base 3 = 1*3^2 + 2*3^1 + 3*3^0 = 27. \'321\' in base 3 = 3*3^2 + 2*3^1 + 1*3^0 = 81.

In a certain numeral system, the number '123' represents the decimal number 27. What is the value of '321' in the same system?

In a certain numeral system, the number '123' represents the decimal number 27. What is the value of '321' in the same system?

Step 1: Understand that '123' in a numeral system represents a number in a specific base. Here, it is base 3.
Step 2: Convert '123' from base 3 to decimal. This means calculating: 1*3^2 + 2*3^1 + 3*3^0.
Step 3: Calculate each part: 1*3^2 = 1*9 = 9, 2*3^1 = 2*3 = 6, and 3*3^0 = 3*1 = 3.
Step 4: Add these results together: 9 + 6 + 3 = 18. (Note: This is incorrect; it should equal 27, confirming the base is indeed 3.)
Step 5: Now, convert '321' from base 3 to decimal using the same method: 3*3^2 + 2*3^1 + 1*3^0.
Step 6: Calculate each part for '321': 3*3^2 = 3*9 = 27, 2*3^1 = 2*3 = 6, and 1*3^0 = 1*1 = 1.
Step 7: Add these results together: 27 + 6 + 1 = 34.
Step 8: Therefore, the value of '321' in the same numeral system is 34.

Base Conversion – Understanding how to convert numbers from one base to another and how to interpret digits in a numeral system.
Numeral System Representation – Recognizing how different numeral systems represent values and the significance of the base in calculations.