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Exponents with Like Bases

Let Pn=n5-n...

Question

Let $P (n) = n^{5} - n$ , then the largest of the following integers which divides $P (1), P (2), \dots P (n)$ is

1

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60

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16

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30

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Solution

The correct option is D $30$
${n^{5} - n = n \times (n}^{4} - 1) = {n \times (n}^{2} - 1) {(n}^{2} + 1) = n \times (n - 1) \times (n + 1) \times {(n}^{2} + 1)$
If $n$ is any natural no. , one of $n, n - 1, n + 1$ will be divisible by $3.$
Also if $n$ is even, then $(n - 1) \times (n + 1) \times {(n}^{2} + 1)$ is odd, therefore $P (n)$ will be divisible by $2$ and not $4.$
Hence, option D is correct

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