111 base 3 to 8

Answer 1

To convert the number \( 111 \) from base 3 to base 8, we can follow these steps:

1. Convert from base 3 to base 10.
2. Convert from base 10 to base 8.

Step 1: Convert from base 3 to base 10

The base 3 number \( 111_3 \) can be expanded in base 10 as follows:

\[ 111_3 = 1 \times 3^2 + 1 \times 3^1 + 1 \times 3^0 \]

Calculating each term:

\[ 1 \times 3^2 = 1 \times 9 = 9 \] \[ 1 \times 3^1 = 1 \times 3 = 3 \] \[ 1 \times 3^0 = 1 \times 1 = 1 \]

Adding these together:

\[ 9 + 3 + 1 = 13 \]

Thus, \( 111_3 = 13_{10} \).

Step 2: Convert from base 10 to base 8

Now, we convert \( 13 \) from base 10 to base 8. We do this by dividing the number by 8 and recording the remainder.

\[ 13 \div 8 = 1 \quad \text{remainder} = 5 \] \[ 1 \div 8 = 0 \quad \text{remainder} = 1 \]

Reading the remainders from bottom to top gives us \( 15_8 \).

Final Result

The base 3 number \( 111_3 \) is equivalent to \( 15_8 \) in base 8.