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n!^{f(n)} divides (n^n)! when f(n) = (n^n - 1)/(n - 1)

Source: Austrian Polish 2003 APMC

April 25, 2020
number theorydividesfactorialdivisible

Problem Statement

Put f(n)=nn−1n−1f(n) = \frac{n^n - 1}{n - 1}. Show that n!f(n)n!^{f(n)} divides (nn)!(n^n)! . Find as many positive integers as possible for which n!f(n)+1n!^{f(n)+1} does not divide (nn)!(n^n)! .
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