102 base 3 to 8

Answer 1

To convert the number \( 102_3 \) (which is in 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 \( 102_3 \) can be converted to base 10 as follows:

\[ 102_3 = 1 \times 3^2 + 0 \times 3^1 + 2 \times 3^0 \]

Calculating each term:

• \( 1 \times 3^2 = 1 \times 9 = 9 \)
• \( 0 \times 3^1 = 0 \)
• \( 2 \times 3^0 = 2 \times 1 = 2 \)

Now, adding these together:

\[ 9 + 0 + 2 = 11 \]

So, \( 102_3 = 11_{10} \).

Step 2: Convert from base 10 to base 8

Now, we convert \( 11_{10} \) to base 8. To do this, we perform division by 8 and keep track of the remainders:

\[ 11 \div 8 = 1 \quad \text{remainder } 3 \]

Since the quotient is 1 and less than 8, we stop here.

The number \( 11_{10} \) can be represented in base 8 as follows:

• The remainders (read from bottom to top) give us \( 13_8 \).

Final Result

Thus, the base 3 number \( 102_3 \) converts to \( 13_8 \) in base 8.

\[ \boxed{13_8} \]