# Multiplicative Functions

Definition 3.1   A function is multiplicative if, whenever and , we have

Recall that the Euler -function is

and

Proposition 3.2   is a multiplicative function.

Proof. Suppose that and . Consider the map

 and and     and

defined by

mod     mod

The map  is injective: If , then and , so, since , , so .

The map  is surjective: Given with , , the Chinese Remainder Theorem implies that there exists with and . We may assume that , ans since and , we must have . Thus .

Because is a bijection, the set on the left has the same size as the product set on the right. Thus

Example 3.3   The proposition makes it easier to compute . For example,

Also, for , we have

since is the number of numbers less than minus the number of those that are divisible by . Thus, e.g.,

The function is also available in PARI:
  ? eulerphi(389*11^2)
%15 = 42680


Question 3.4   Is computing 1000 digit number really easy or really hard?