Is the group of positive rationals under multiplication isomorphic to its subgroup consisting of rationals with odd numerators and denominators?

This is a TIFR entrance exam question. I have spent much time on it but not able to come to conclusion.

We have given the group of positive rationals with multiplication and it's subgroup of those rationals whose numerator and denominator are both odd positive integers.

I saw here that both of the above groups are isomorphic but I'm really unfamiliar to such an approach.

I have gone through Group Theory chapter of Gallian but still not finding any way.

Hint: $$ \mathbf {Q^{+}}^{\times} \cong \bigoplus_{p} \mathbf Z $$

where the direct sum has one copy of $ \mathbf Z $ for each prime $ p \in \mathbf N $. (What is an explicit isomorphism? How does this help you solve your problem?)

Let $G$ denote the subgroup of $(\mathbb{Q},.) $ with odd numerator and denominator. Let $(p_n)$ be sequence of primes that is $p_1=2,p_2=3, \ldots$. Now define $\phi:(\mathbb{Q},.) \rightarrow G $ by $\phi(p_i)=p_{i+1}$ and extend. We can do that because primes generate them freely. This can be verified to be a isomorphism.