# 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, 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 $b/a$.