Divisibility

Task number: 3314

Prove by mathematical induction that \(4 |(6n^2+2n)\).

  • Solution

    For \(n=1\):
    \(6{\cdot} 1^2+2{\cdot} 1=8\), so \(4 |(6{\cdot} 1^2+2{\cdot} 1)\)

    Induction step \(n \to n+1\):
    \(6(n+1)^2+2(n+1)=6n^2+12n+6+2n+2=(6n^2+2n)+4(3n+2)\).

    Because \(4 |(6n^2+2n)\) and also \(4 |4(3n+2)\), we have \(4 |6(n+1)^2+2(n+1)\).

Difficulty level: Easy task (using definitions and simple reasoning)
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