To convert the number \(112\) from base \(3\) to base \(8\), we can first convert it from base \(3\) to base \(10\) (decimal), and then from base \(10\) to base \(8\).
Step 1: Convert from base 3 to base 10
The number \(112\) in base \(3\) can be expanded as follows:
\[ 1 \times 3^2 + 1 \times 3^1 + 2 \times 3^0 \]
Calculating that:
\[ 1 \times 9 + 1 \times 3 + 2 \times 1 = 9 + 3 + 2 = 14 \]
So, \(112\) base \(3\) is equal to \(14\) in base \(10\).
Step 2: Convert from base 10 to base 8
Now, we will convert \(14\) from base \(10\) to base \(8\). We can do this by repeatedly dividing by \(8\) and keeping track of the remainders.
- \(14 \div 8 = 1\) with a remainder of \(6\).
- \(1 \div 8 = 0\) with a remainder of \(1\).
Now we read the remainders from bottom to top:
The base \(8\) representation of \(14\) is \(16\).
Conclusion
Thus, \(112\) in base \(3\) is equal to \(16\) in base \(8\).
\[ \text{Answer: } 16 \text{ (base 8)} \]
1 answer