The largest of the following integers which divides each of the members of the sequence $1^{5} - 1, 2^{5} - 2, 3^{5} - 3, . . . ., n^{5} - n, . . .$ is:

1

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60

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15

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30

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Solution

The correct option is B $30$
$S e q u e n c e : 1^{5} - 1, 2^{5} - 2, . . . . ., n^{5} - n, . . . . n^{5} - n = n (n^{4} - 1) = n (n - 1) (n + 1) (n^{2} + 1) n = 1, 2, 3, . . . . .$
$A t n = 1, n^{5} - n = 0$ [Not Possible]
$A t n = 2, n^{5} - n = 30$
$∴$ Each number in the sequence is a multiple of 30.
$∴$ Each number is divisible by 30.

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