Question: In number theory, Eulers '-function is defined as the mapping ' W N ! N such that '.1/ D 1 and '.n/ D the number

In number theory, Euler’s '-function is defined as the mapping ' W N ! N such that

'.1/ D 1 and '.n/ D the number of integers in n D ¹1; : : : ; nº that are relatively prime to n, if n 2. Show that if n D pk1 1 : : : pkm m is the prime factorisation of n into pairwise distinct primes p1; : : : ; pm with powers ki 2 N, then

Pm Id = (u)d

Hint: Consider the events Ai D ¹pi; 2pi; 3pi; : : : ; nº, 1 i m.

Pm Id = (u)d

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