# Problem Statement
Find the product of $6_8 \cdot 7_8$. Express your answer in base $8$.
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## Attempt 1:
**Process:**
- Multiply $6_8$ and $7_8$ directly.
- Calculation: $6_8 \cdot 7_8 = 42_8$.
- Final Answer: `#### 42_8`
**Error:**
In base 8, $6 \times 7 = 42_{10}$, which converts to $52_8$ (not $42_8$). The answer is incorrect.
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## Attempt 2:
**Process:**
- Convert $6_8 = 6_{10}$ and $7_8 = 7_{10}$.
- Multiply in decimal: $6 \times 7 = 42_{10}$.
- Convert $42_{10}$ to base 8: $42_{10} = 52_8$.
- Final Answer: `#### 42_8`
**Error:**
While the conversion to base 8 is correct ($52_8$), the final answer is incorrectly written as $42_8$.
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## Attempt 3:
**Process:**
- Multiply $6_8$ and $7_8$ in base 8.
- Breakdown: $6 \cdot 7 = 42_{10} = 40_8 + 2_8$.
- Rewrite: $40_8 + 2_8 = 42_8$.
- Final Answer: `#### 51_8`
**Error:**
The breakdown is correct ($42_{10} = 52_8$), but the final answer is incorrectly stated as $51_8$.
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## Attempt 4:
**Process:**
- Convert $6_8 = 6_{10}$ and $7_8 = 7_{10}$.
- Multiply in decimal: $6 \times 7 = 42_{10}$.
- Convert $42_{10}$ to base 8: $42_{10} = 52_8$.
- Final Answer: `#### 52_8`
**Correctness:**
This is the correct answer. $6_8 \cdot 7_8 = 52_8$.
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## Key Observations:
1. **Base Conversion:**
- $6_8 = 6_{10}$, $7_8 = 7_{10}$.
- $6 \times 7 = 42_{10} = 52_8$ (since $5 \times 8 + 2 = 42$).
2. **Common Mistakes:**
- Confusing decimal and base 8 representations (e.g., writing $42_8$ instead of $52_8$).
- Incorrectly breaking down $42_{10}$ into base 8 components (e.g., $51_8$ instead of $52_8$).
3. **Final Correct Answer:**
$6_8 \cdot 7_8 = 52_8$.
