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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