= Solution
A <Non-Archimedean absolute value> on a field is a function $|\cdot|:K\to\mathbb R_{\ge0}$ with $|x|=0$ exactly for $x=0$, $|xy|=|x||y|$, and the <ultrametric inequality> $|x+y|\le\max(|x|,|y|)$. It is nontrivial if some nonzero element has value different from one. <Equivalent absolute values> induce the same topology; for nontrivial absolute values this is equivalent to $|x|_2=|x|_1^c$ for some $c>0$.
Here is also a proof of the latter equivalence in the case needed. A positive power plainly preserves the open neighborhoods of zero. Conversely, if the topologies coincide, then $|x|_1<1$ if and only if $x^n\to0$ in the first topology, hence if and only if $|x|_2<1$. Choose $a$ with $0<|a|_1<1$. For every $x\ne0$, $m\in\mathbb Z$ and $n\ge1$, whether $|x^n/a^m|<1$ agrees for the two values. Comparison with all rational numbers $m/n$ therefore identifies the two ratios $\log|x|_j/\log|a|_j$. Thus $|x|_2=|x|_1^c$ with $c=\log|a|_2/\log|a|_1>0$.
Now let $|\cdot|$ be nontrivial and non-Archimedean on $\mathbb Q$. Since $|1|=1$ and every positive integer is a sum of ones, $|n|\le1$ for every integer. If all nonzero integers had value one, multiplicativity would make the absolute value trivial on all rationals. Hence some integer has value less than one; its prime factorization shows that some prime $p$ has $|p|<1$.
There cannot be two distinct such primes $p,q$. A <Bezout identity> $ap+bq=1$ with $a,b\in\mathbb Z$ would give
$$
1=|ap+bq|\le\max(|a||p|,|b||q|)\le\max(|p|,|q|)<1.
$$
Thus every other prime has value one. Prime factorization of the numerator and denominator of $x\in\mathbb Q^\times$ now gives
$$
|x|=|p|^{v_p(x)}=p^{-c v_p(x)}=|x|_p^c,\qquad c=-\frac{\log|p|}{\log p}>0.
$$
Conversely $|x|_p=p^{-v_p(x)}$ is multiplicative and satisfies the <ultrametric inequality>, since $v_p(x+y)\ge\min(v_p(x),v_p(y))$. Distinct primes give inequivalent values: $p^m\to0$ for the $p$-adic absolute value but $|p^m|_q=1$ for $q\ne p$. We have proved the <non-Archimedean part of Ostrowski theorem>:
$$
\boxed{\text{The nontrivial non-Archimedean absolute values on }\mathbb Q\text{ are }|\cdot|_p^c,\quad p\text{ prime},\ c>0.}
$$
There is exactly one equivalence class for each prime.
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