Euler's phi function

Question

I have attempted a problem which required me to use Euler's phi function. In doing so I have assumed that $\varphi(xy)=\varphi(x)\varphi(y).$

Am I right to do this or have I made a mistake?

Phi function is multiplicative only when x & y are relative primes. — Subhadeep Sarkar, May 01 '21 at 07:15

score 3 · Accepted Answer · edited Apr 13 '17 at 12:19

3

You can use this only if $x,y$ are coprime.

E.g. see this post:

edited Apr 13 '17 at 12:19

Community

answered Mar 14 '16 at 21:02

flawr

score 1 · Answer 2 · answered Mar 14 '16 at 21:07

1

If $x$ and $y$ are relatively prime, that is, they have no common divisors, then this is true. However, in general it is false.

For example, for $x = y = 3$, we have that $\phi(x) = \phi(y) = \phi(3) = 2$ but $\phi(xy) = \phi(9) = 6$.

answered Mar 14 '16 at 21:07

John Doe

score 0 · Answer 3 · 2021-05-01T08:14:09.830

0

The situation is expressed by the language that $\varphi $ is multiplicative, but not completely multiplicative.

It is essentially the Chinese remainder theorem, which tells us that when $(x,y)=1$, we have $\varphi (xy)=\varphi (x)\varphi (y) $.

(It is not always true without the restriction that $x $ and $y $ are co-prime.)

edited May 01 '21 at 08:14

answered May 01 '21 at 08:03

3 Answers3