Convert 121 base 3 to 8

Answer 1

To convert the number \(121\) from base \(3\) to base \(8\), we can follow these steps:

1. Convert from base \(3\) to base \(10\) (decimal).
2. Convert from base \(10\) to base \(8\).

Step 1: Convert \(121_3\) to base \(10\)

The base \(3\) number \(121_3\) can be expanded using powers of \(3\): \[ 121_3 = 1 \cdot 3^2 + 2 \cdot 3^1 + 1 \cdot 3^0 \] Calculating each term: \[ 1 \cdot 3^2 = 1 \cdot 9 = 9 \] \[ 2 \cdot 3^1 = 2 \cdot 3 = 6 \] \[ 1 \cdot 3^0 = 1 \cdot 1 = 1 \]

Adding these values together: \[ 9 + 6 + 1 = 16 \]

So, \(121_3\) is equal to \(16_{10}\).

Step 2: Convert \(16_{10}\) to base \(8\)

Now we will convert \(16\) from base \(10\) to base \(8\). We do this by dividing the number by \(8\) and recording the remainder.

1. Divide \(16\) by \(8\): \[ 16 \div 8 = 2 \quad \text{remainder = } 0 \]

2. Now, take the quotient \(2\) and divide by \(8\): \[ 2 \div 8 = 0 \quad \text{remainder = } 2 \]

Now, we read the remainders from bottom to top, which gives us \(20_8\).

Final Answer

Thus, the base \(8\) equivalent of \(121_3\) is: \[ \boxed{20_8} \]