To convert the number \(101_3\) (base 3) to base 8, we first need to convert it to base 10 (decimal), and then from base 10 to base 8.
Step 1: Convert from base 3 to base 10
The base 3 number \(101_3\) can be converted to decimal using the formula:
\[ (1 \times 3^2) + (0 \times 3^1) + (1 \times 3^0) \]
Calculating it step by step:
\[ 1 \times 3^2 = 1 \times 9 = 9 \] \[ 0 \times 3^1 = 0 \times 3 = 0 \] \[ 1 \times 3^0 = 1 \times 1 = 1 \]
Now, add these results together:
\[ 9 + 0 + 1 = 10 \]
So, \(101_3 = 10_{10}\) in decimal.
Step 2: Convert from base 10 to base 8
Now, we convert \(10_{10}\) to base 8 by division by 8 and keeping track of remainders:
- \(10 \div 8 = 1\) remainder \(2\)
- \(1 \div 8 = 0\) remainder \(1\)
We stop since the quotient is now zero. Reading the remainders from bottom to top gives us \(12\).
Thus, \(10_{10} = 12_8\) in base 8.
Final Result
The final conversion of \(101_3\) to base 8 is:
\[ 101_3 = 12_8 \]
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