euler func

**anncar** · December 8th 2007, 08:53 PM

Prove that phee(n) is even for any n greater than or equal to 3 (here phee(n) is the Euler phee-
function).

**Constatine11** · December 9th 2007, 07:43 AM

Originally Posted by anncar

Prove that phee(n) is even for any n greater than or equal to 3 (here phee(n) is the Euler phee-
function).

What do you know about $\phi$ ?

If you know that for $n=p_1^{k_1}.. p_r^{k_r}$ , then:

$\phi(n)=(p_1-1)p^{k_1-1}..(p_r-1)p^{k_r-1}.$

And it follows immeadiatly that if $n>2$ that $\phi$ is even (as if it has an odd prime factor $p_i$ , then $(p_i-1)$ is even and so $\phi(n)$ is even, or if it only has even prime factors then it is a power k of 2 greater than the first and $2^{k-1}$ is even)

ZB

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