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Exponent of five in a certain expression

Source: 2018 China Southeast MO Grade 10 P8

July 31, 2018

number theory

Problem Statement

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.