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I'm inclined to think yes because marginal cost only depends on variable cost (fixed costs don't matter), but I'm not 100% certain.
Basically, my thought process is that marginal cost of producing one additional unit is the change in total variable cost to produce that unit. So, my current idea is that if marginal cost is constant, that must mean that average variable cost (total variable cost/output) is also constant. But I'm still not entirely convinced.
Your intuition is correct.
First, you're right that "marginal cost only depends on variable cost", since \begin{equation} MC(q)=\frac{\mathrm dTC(q)}{\mathrm dq}=\frac{\mathrm d(FC+VC(q))}{\mathrm dq}=\frac{\mathrm dVC(q)}{\mathrm dq}. \end{equation} Next, if marginal cost is some constant $k$, then variable cost must be $VC(q)=kq$, because we can integrate $MC$ to obtain $TC$, where the term that varies with $q$ is $kq$: \begin{equation} MC(q)=k \quad\Rightarrow\quad \int k\;\mathrm dq =\underbrace{kq}_{VC(q)} +\underbrace{K\vphantom{q}}_{FC} \end{equation} where $K$ is a constant resulting from the integration. Lastly, \begin{equation} AVC(q)=\frac{VC(q)}{q}=\frac{kq}{q}=k, \end{equation} which is a constant, and same as $MC$.
