Problem
NT-B2-M06-P012 Divisibility by \(16\)
#12
★★★★☆ Level 4 of 5
Find all \(n\) such that \(16\mid3^n-1\).
Split by parity of \(n\).
If \(n\) is odd, \(v_2(3^n-1)=v_2(2)=1\), not enough. If \(n\) is even, then \(v_2(3^n-1)=v_2(2)+v_2(4)+v_2(n)-1=2+v_2(n)\). We need \(2+v_2(n)\ge4\), so \(v_2(n)\ge2\). Answer: \(4\mid n\).
Example of a find-all-\(n\) LTE problem.