Problem

Source: 2018 China Southeast MO Grade 10 P8

Tags: number theory



Given a positive integer $m$. Let $$A_l = (4l+1)(4l+2)...(4(5^m+1)l)$$for any positive integer $l$. Prove that there exist infinite number of positive integer $l$ which $$5^{5^ml}\mid A_l\text{ and } 5^{5^ml+1}\nmid A_l$$and find the minimum value of $l$ satisfying the above condition.