[117 points] Compute $$ 234_{7} \cdot 567 $$. Express your answer in base 7 .

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To compute $$ 234_{7} \cdot 567 $$ and express the answer in base 7, follow these steps:

1. **Convert $$ 234_{7} $$ to Base 10:**
   $$
   234_7 = 2 \times 7^2 + 3 \times 7^1 + 4 \times 7^0 = 2 \times 49 + 3 \times 7 + 4 \times 1 = 98 + 21 + 4 = 123_{10}
   $$

2. **Multiply by 567 in Base 10:**
   $$
   123_{10} \times 567_{10} = 69,741_{10}
   $$

3. **Convert 69,741 from Base 10 to Base 7:**
   - Find the highest power of 7 less than or equal to 69,741, which is $$ 7^5 = 16,807 $$.
   - **Divide and find coefficients:**
     $$
     \begin{align*}
     69,741 \div 16,807 &= 4 \quad \text{(remainder: } 2,513) \
     2,513 \div 2,401 &= 1 \quad \text{(remainder: } 112) \
     112 \div 343 &= 0 \quad \text{(remainder: } 112) \
     112 \div 49 &= 2 \quad \text{(remainder: } 14) \
     14 \div 7 &= 2 \quad \text{(remainder: } 0) \
     0 \div 1 &= 0
     \end{align*}
     $$
   - **Compile the coefficients:**
     $$
     7^5: 4,\ 7^4: 1,\ 7^3: 0,\ 7^2: 2,\ 7^1: 2,\ 7^0: 0
     $$
   - **Write the number in base 7:**
     $$
     410220_7
     $$

**Final Answer:**
$$
410220_{7}
$$
The product of $$ 234_{7} $$ and 567 in base 7 is $$ 410220_{7} $$.

[117 points] Compute \( 234_{7} \cdot 567 \). Express your answer in base 7 .

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