I think the solution in the editorial is overkill. My solution:

• You want to write all values as products of integers in $$$[i, j]$$$, where $$$j-i$$$ is the smallest possible.
• For each $$$i$$$, you want to find the smallest $$$j$$$. If $$$i$$$ increases, $$$j$$$ increases, so you can use two pointers and try to answer "can you write all values only using $$$[i, j]$$$?". After processing $$$[i, j]$$$, you must be able to process either $$$[i, j+1]$$$ or $$$[i+1, j]$$$ fast.
• Let $$$f(x)$$$ be $$$1$$$ if you can write $$$x$$$ using $$$[i, j]$$$, $$$0$$$ otherwise. The transition from $$$[i, j]$$$ to $$$[i, j+1]$$$ is $$$f(k(j+1))$$$ |= $$$f(k)$$$ for all $$$k$$$ from $$$1$$$ to $$$m/(j+1)$$$ (similar to adding $$$j+1$$$ to a knapsack with product instead of sum).
• Removing $$$i$$$ from the knapsack is possible if you let $$$f(x)$$$ be the number of ways to write $$$x$$$, modulo a random prime (instead of $$$0$$$ or $$$1$$$). For example, see abc321_f.

The complexity is $$$O(n + m \log m)$$$, because each $$$i$$$ is added and removed from the knapsack at most once.