On Euler's Φ-Function and its Extensions

The type of argument given in text-books to derive the form of Φ(n)-the number of numbers less than n and prime to n, is as follows. If n be the given number and p₁, p₂,...,p_q be the different prime factors of n, then there are n/Pi numbers divisible by pi, n/pj numbers divisible by pj, and so on. Generally there are n/pipjpk..... numbers divisible by pi, pj, pk,......, simultaneously.