Problem
NT-B2-M06-P017 When \(3^n\) Divides
#17
★★★★★ Level 5 of 5
Find all positive integers \(n\) such that \(3^n\mid10^n-1\).
Compare \(n\) with \(v_3(10^n-1)\).
By LTE, \(v_3(10^n-1)=v_3(9)+v_3(n)=2+v_3(n)\). We need \(2+v_3(n)\ge n\). The values \(n=1,2,3\) work. If \(n\ge4\), then \(2+v_3(n)
Strong problem: after LTE, an inequality remains.