Problem
NT-B2-M06-P015 Powers of Three
#15
★★★★☆ Level 4 of 5
Find the largest \(k\) such that \(3^k\mid2^{2n}-1\).
Replace \(2^{2n}\) by \(4^n\).
\(2^{2n}-1=4^n-1\). Since \(3\mid4-1\), \(v_3(4^n-1)=1+v_3(n)\). Therefore \(k=1+v_3(n)\).
A hidden base change makes the problem direct.