If $\frac{\phi(n)}{n}=\frac{\phi(m)}{m}$ then $n$ and $m$ have the same primes in their prime factorization

Question

Here $m,n\in\mathbb{N}\backslash \{0\}$ and $\phi$ is the Euler Totient function. I have been trying to prove that this statement is true but I'm stuck. I started with applying the product formula for $\phi$ to get that the fraction becomes $\prod_{p\mid n} (1-\frac{1}{p})=\prod_{q\mid m} (1-\frac{1}{q})$, where $p$ and $q$ are primes. I then did some cross multiplication to obtain that: $$\left(\prod_{q\mid m} (q-1)\right)\cdot\left(\prod_{p\mid n} p\right)=\left(\prod_{p\mid n} (p-1)\right)\cdot\left(\prod_{q\mid m} q\right)$$ Now we know that the largest of the $p_{i}=p_{l}$ terms do not divide any of the $p-1$ terms so therefore there is a $q_{k}=q_{r}$ that it divides instead, and since both numbers are prime $p_{l}=q_{r}$. If we continue this process we find that each $p_{i}$ is equal to some $q_{k}$. Now over here is where I got stuck because even though each $p_{i}$ is equal to some $q_{k}$ there may be more $q_{k}$ left over and, so $m$ and $n$ may not have all of the same prime factors. I feel I may be missing something or have done something incorrectly, so any advice would be helpful (thank you in advance).

Answer 1

You are there, you just have to explain your solution better.

Let $p_1<p_2<...<p_k$ be the distinct primes dividing $n$ and $q_1<q_2<...<q_j$ be the distinct primes dividing $m$.

You used $$\prod_{i=1}^k \left(1-\frac{1}{p_i} \right)=\prod_{i=1}^j \left(1-\frac{1}{q_i} \right) (*)$$ to conclude that $p_k=q_j$. Now, forget about $m,n$.

Using $p_k=q_j$ in $(*)$ you get: $$\prod_{i=1}^{k-1} \left(1-\frac{1}{p_i} \right)=\prod_{i=1}^{j-1} \left(1-\frac{1}{q_i} \right) $$

And repeat the process.

To make it a formal proof, prove by induction on $k$ the following Lemma:

Lemma If $p_1<p_2<...<p_k$ and $q_1<q_2<...<q_j$ are prime numbers such that
$$\prod_{i=1}^k \left(1-\frac{1}{p_i} \right)=\prod_{i=1}^j \left(1-\frac{1}{q_i} \right) $$ then $k=j$ and $$p_1=q_1; p_2=q_2;...;p_k=q_k \,.$$

Answer 2

Dividing out the common factors, we have $$ \prod_{p \in A} \frac{p-1}{p} = \prod_{p \in B} \frac{p-1}{p} $$ where $A$ is the set of primes dividing $m$ but not $n$, and $B$ the set of primes dividing $n$ but not $m$. Suppose these sets are not both empty. Then there is a largest prime in $A \cup B$. The denominator of one side is divisible by it, but the denominator of the other side is not.