Products of consecutive integers dividing one another

I’ve been working on a project lately and I find myself stuck with the following problem.

Problem. Given positive integers a<b, what can be said about the least positive integer t such that a(a+1)\cdots (a+t) does not divide b(b-1)\cdots (b-t)?

Obviously t is bounded above by b-a+1 but this is very weak. Probably the least t should be close to b/a.

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