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2

Arithmetical functions

2.1 The function [x]

For any real x, one signifies by [x] the largest integer ≤ x, that is, the unique integer such that x − 1 < [x] ≤ x. The function is called ‘the integral part of x’. It is readily verified that [x + y] ≥ [x] + [y] and that, for any positive integer n, [x + n] = [x] + n and [x/n] = [[x]/n]. The difference x − [x] is called ‘the fractional part of x’; it is written {x} and satisfies 0 ≤ {x} < 1.

Let now p be a prime. The largest integer l such that pl divides n! can be neatly expressed in terms of the above function. In fact, on noting that [n/p] of the numbers 1, 2, …, n are divisible by p, that [n/p2] are divisible by p2, and so on, we obtain

It follows easily that l ≤ [n/(p − 1)]; for the latter ...

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