Problem
NT-B2-M05-P020 Impossible Divisibility
#20
★★★★★ Level 5 of 5
Prove that \(n!\) is not divisible by \(3^n\) for any positive integer \(n\).
Bound \(v_3(n!)\) above by \(n/3+n/9+\cdots\).
By Legendre, \(v_3(n!)=\lfloor n/3\rfloor+\lfloor n/9\rfloor+\cdots\). This sum is strictly less than \(n/3+n/9+\cdots=n/2\), hence certainly less than \(n\). Therefore \(v_3(n!)
Closes the module with a valuation estimate without exact computation.