11th Class Mathematics Other Series Question Bank Critical Thinking

  • question_answer Let \[n(>1)\] be a positive integer, then the largest integer \[m\] such that \[({{n}^{m}}+1)\] divides \[(1+n+{{n}^{2}}+.......+{{n}^{127}})\], is [IIT 1995]

    A) 32

    B) 63

    C) 64

    D) 127

    Correct Answer: C

    Solution :

    Since \[{{n}^{m}}+1\] divides \[1+n+{{n}^{2}}+.......+{{n}^{127}}\] Therefore \[\frac{1+n+{{n}^{2}}+......+{{n}^{127}}}{{{n}^{m}}+1}\] is an integer \[\Rightarrow \] \[\frac{1-{{n}^{128}}}{1-n}\times \frac{1}{{{n}^{m}}+1}\] is an integer \[\Rightarrow \] \[\frac{(1-{{n}^{64}})(1+{{n}^{64}})}{(1-n)({{n}^{m}}+1)}\]  is an integer when largest \[m=64\].

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