Variant of Bounded Subset Product

Consider the following decision problem:

Given $$([b_1, \cdots, b_n],t)$$, where $$[b_1, \cdots b_n]$$ is an array of natural numbers each less than $$n^c$$ and $$t$$ is a target natural number, do there exist natural numbers $$f_1, \cdots f_n$$ such that

1. $$f_i \le b_i$$ for all $$1 \le i \le n$$
2. $$\prod_{i=1}^{n} f_i = t$$?

We guess that if the prime factorization of the target $$t$$ is given, then the problem might be polynomial, and in general, it seems similar to subset-product and might be NP-complete, but we couldn't prove an exact result or find a reference.

Are there any references/results for time complexity of this problem (in the case where the prime factorization of $$t$$ is given and when it isn't)? If an unpublished solution exists, we'd appreciate a hint!

• I think it is unlikely that the problem is in P if the factorization of $t$ is given but NP-complete if the factorization is not given, as factoring is considered unlikely to be NP-complete.
– D.W.
Commented Dec 11, 2023 at 0:05
• Since each number $b_i$ is less than $n^c$ and the input size is proportional to $n$, using a naive prime factorization algorithm for $t$, keeps the running time polynomial. So, it is safe to assume that prime factorization is given for $t$. Commented Dec 11, 2023 at 9:44