5. [126 points] Convert $$ 333_{10} $$ to base 6.

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UpStudy AI · Accepted Answer

To convert the decimal number $$ 333_{10} $$ to base 6, follow these steps:

1. **Divide the number by 6 and keep track of the remainders.**

$$
   \begin{align*}
   333 \div 6 &= 55 \quad \text{with a remainder of } 3 \
   55 \div 6 &= 9 \quad \text{with a remainder of } 1 \
   9 \div 6 &= 1 \quad \text{with a remainder of } 3 \
   1 \div 6 &= 0 \quad \text{with a remainder of } 1 \
   \end{align*}
   $$

2. **List the remainders from the last division to the first.**

The remainders, read from bottom to top, are: **1**, **3**, **1**, **3**

3. **Combine the remainders to form the base 6 number.**

So, $$ 333_{10} $$ in base 6 is $$ 1313_6 $$.

4. **Verification:**

To ensure the conversion is correct, you can convert $$ 1313_6 $$ back to decimal:

$$
   1 \times 6^3 + 3 \times 6^2 + 1 \times 6^1 + 3 \times 6^0 = 1 \times 216 + 3 \times 36 + 1 \times 6 + 3 \times 1 = 216 + 108 + 6 + 3 = 333
   $$

The conversion is verified as correct.

**Final Answer:**

$$
333_{10} = 1313_6
$$
$$ 333_{10} = 1313_6 $$

5. [126 points] Convert \( 333_{10} \) to base 6.

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