\5_6 \overline{) 3410_6}\

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Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Base Conversion", "Base Arithmetic" ], "new_concepts": [ "Base Division" ], "current_concepts": [ "Base Conversion", "Base Arithmetic", "Base Division" ] } ## Convert to base 10 Using the *Base Conversion* knowledge point $$ \begin{aligned} &3410_6 = 3 \times 6^3 + 4 \times 6^2 + 1 \times 6^1 + 0 \times 6^0\ &3410_6 = 3 \times 218 + 4 \times 36 + 6 + 0\ &3410_6 = 648 + 144 + 6 = 798_{10}\ &5_6 = 5_{10} \end{aligned} $$ ## Perform division in base 10 Using the *Base Arithmetic* knowledge point $$ \begin{aligned} &798 \div 5 = 159 \text{ R } 3 \end{aligned} $$ ## Convert quotient and remainder back to base 6 Using the *Base Conversion* knowledge point $$ \begin{aligned} &159 \div 6 = 26 \text{ R } 3\ &26 \div 6 = 4 \text{ R } 2\ &4 \div 6 = 0 \text{ R } 4\ &159_{10} = 423_6\ &3_{10} = 3_6 \end{aligned} $$ ## Verify using base 6 long division We perform long division directly in base 6: - $34_6 \div 5_6$: Since $5_6 \times 4_6 = 32_6$, the first digit is $4_6$. - Subtract: $34_6 - 32_6 = 2_6$. Bring down 1 to get $21_6$. - $21_6 \div 5_6$: Since $5_6 \times 2_6 = 14_6$, the second digit is $2_6$. - Subtract: $21_6 - 14_6 = 3_6$. Bring down 0 to get $30_6$. - $30_6 \div 5_6$: Since $5_6 \times 3_6 = 23_6$, the third digit is $3_6$. - Subtract: $30_6 - 23_6 = 3_6$. - The quotient is $423_6$ and the remainder is $3_6$. $3410_6 \div 5_6 =$ $423_6 \text{ R } 3_6$ { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Number Theory", "Base Division" ] }

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Question

Explanation:

Convert to base 10

Perform division in base 10

Convert quotient and remainder back to base 6

Verify using base 6 long division

Answer: