Question about the totient function and congruence classes

Question

If $x,y$ are integers where $x | \varphi(y)$ does it follow that the reduced residue class modulo $y$ divides evenly into congruence classes modulo $x$?

For example, if we look at $y=35$ and $x = 3$. In this case, we have $\varphi(35)=24$ and we see that there are:

• $8$ elements $\{3, 6, 9, 12, 18, 24, 27, 33\}$ that are congruent to $0$ modulo $3$
• $8$ elements $\{ 1, 4, 13, 16, 19, 22, 31, 34\}$ that are congruent to $1$ modulo $3$
• $8$ elements $\{ 2, 8, 11, 17, 23, 26, 29, 32\}$ that are congruent to $2$ modulo $3$

Is this always the case? If not, can you provide a counter example?

Thanks very much!

-Larry

Answer 1

$2|\varphi(12)=4$, but of course all four elements are odd.