Math Is Fun Forum

Fred245

Guest

General number theory question

I noticed that the value of φ(n) is always even whenever n is greater than or equal to 3.  Why is this?

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: General number theory question

First, note that if n is even and n > 2, then

Next, note that

Hence

and for n > 2

Thus, for n > 2, whenever a is coprime with n, so is n−a ≠ a; therefore the integers less than and coprime with n come in distinct pairs — i.e. there are an even number of them.