I'm beginning some self-study in Number Theory and have come across a problem that I'm not really sure how to solve. Here's the problem:
Prove that, if,
$$ a=q_{1}^{e_{1}}q_{2}^{e_{2}} . . . q_{r}^{e_{r}} $$ $$ b=s_{1}^{f_{1}}s_{2}^{f_{2}} . . . s_{u}^{f_{u}} $$
are the factorizations of $a$ and $b$ into primes, where $q_{1}<q_{2}<...<q_{r}$ and $ s_{1}<s_{2}<...<s_{u}$, then there exist primes $t_{1}<t_{2}< . . . <t_{v}$ and non-negative integers $q_{i}$ and $h_{i}$ such that:
$$ a=t_{1}^{h_{1}}t_{2}^{h_{2}} . . . t_{v}^{h_{v}} $$ $$ b=t_{1}^{q_{1}}t_{2}^{q_{2}} . . . t_{v}^{q_{v}} $$
As someone who is relatively new to studying mathematics, I normally try to explain what the problem means in words when I begin trying to solve it. The way I understand it, this problem is asking you to prove that any two numbers can be represented as powers of the same number of primes, namely, $v$.
Intuitively, this makes sense. The way I justified it to myself is as follows... some massive prime, $P$, and 3 can be represented this way:
$$ 3=(2^0)(3^1)(5^0) . . . (P^0) $$ $$ P=(2^0)(3^0)(5^0) . . . (P^1) $$
Where, $v=\pi(P)$. Thus, no matter how big the difference between $a$ and $b$, it would still hold.
My problem is that I'm not really sure where to begin with constructing a proof, so I'm asking for some guidance or advice.
Thanks for any help you can give me.