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The largest value of the positive integer k for which `n^(k)+1` divides `1+n+n^(2)+ . . . .+n^(127)`, is

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The largest value of the positive integer k for which `n^(k)+1` divides `1+n+n^(2)+ . . . .+n^(127)`, is
A. 8
B. 16
C. 32
D. 64
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Correct Answer - D
We have,
`1+n+n^(2)+ . .. . +n^(127)=(n^(128)-1)/(n-1)`
`=((n^(64)-1)(n^(64)+1))/(n-1)`
`=(1+n+n^(2)+ . . .+n^(63))(n^(64)+1)`
Thus, the largest value of k for which `n^(k)+1` divides `1+n+n^(2)+ . . .+n^(127)" is "64`.
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