Find the largest integer which divides every member of the following sequence: $$1^5-1,\ 2^5-2,\ 3^5-3,\cdots\ n^5-n.$$

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Although number theory - the study of the natural numbers - does not typically feature in school curricula it plays a leading role in university at first year and beyond. Having a good grasp of the fundamentals of number theory is useful across all disciplines of mathematics. Moreover, problems in number theory are a great leisure past time as many require only minimal knowledge of mathematical 'content'.