Euler's function Phi

Question

Which of the following statement is/are true?

1. $\phi$$(n)$ is even as many times as it is odd.

2. $\phi$$(n)$ is odd for only two values of $n$.

3. $\phi$$(n)$ is even when $n>2$.

4. $\phi$$(n)$ is odd when $n=2$ or $n$ is odd.

Answer 1

Let $n\ge 3$. We show that $\varphi(n)$ is even. Recall that $\varphi(n)$ is the number of integers in the interval $1\le k\le n-1$ such that $\gcd(k,n)=1$.

Note also that if $n\gt 2$ is even, then $\gcd(n/2,n)=n/2\gt 1$.

Now call two integers $a$ and $b$ in our interval a couple if $a+b=n$. Note that if $a$ and $b$ are a couple, then $\gcd(a,n)=1$ if and only if $\gcd(b,n)=1$.

It follows that if $n\ge 3$, then the integers in the interval $1\le k\le n-1$ such that $\gcd(k,n)=1$ are divided into couples. It follows that if $n\ge 3$ then $\varphi(n)$ is even.

This should provide enough information to answer the question. For completeness, note that $\varphi(1)=\varphi(2)=1$.

Answer 2

φ(n) equals the number of distinct generators of a cyclic group G of order n. If n is > 2, and g is a generator of G, then g^{-1} is also a generator of G different from g, because n > 2. So the generators appear in pairs and 2 divides φ(n).