Possible Duplicate:
Why is $f(x) = x\phi(x)$ one-to-one?
How would one show $x \phi (x) = y\phi (y)$ implies $x=y$, where $\phi$ is the Euler totient function?
I have thought of the obvious method of writing $x$ and $y$ in their prime factorization:
$x=p_1^{a_1}p_2^{a_2}\cdots p_k^{a_k}$ (primes arranged in ascending order)
$y=q_1^{b_1}q_2^{b_2}\cdots q_l^{b_l}$ (primes arranged in ascending order)
Then, $x \phi (x)=p_1^{a_1}p_2^{a_2}\cdots p_k^{a_k}(p_1^{a_1}-p_1^{a_1-1})(p_2^{a_2}-p_2^{a_2-2})\cdots (p_k^{a_k}-p_k^{a_k-1})$
$y \phi (y)=q_1^{b_1}q_2^{b_2}\cdots q_k^{b_k}(q_1^{b_1}-q_1^{b_1-1})(q_2^{b_2}-q_2^{b_2-2})\cdots (q_l^{b_l}-q_l^{b_l-1})$
A hint in the book told us to prove that $p_k^{a_k}$ divides $y$. However, I do not have any idea how to do that. Also, even if I managed to prove that, I don't see how that may lead to the final result.
Sincere thanks for any help!