MathDB

NT2

Part of 2015 JBMO Shortlist

Problems(1)

exists k such 41 / repunit with n digits iff k / n (also 37 / n repunit iff 3/n)

Source: JBMO Shortlist NT2

4/24/2019
A positive integer is called a repunit, if it is written only by ones. The repunit with nn digits will be denoted as 11⋯1⏟n\underbrace{{11\cdots1}}_{n} . Prove that: α) the repunit 11⋯1⏟n\underbrace{{11\cdots1}}_{n}is divisible by 3737 if and only if nn is divisible by 33 b) there exists a positive integer kk such that the repunit 11⋯1⏟n\underbrace{{11\cdots1}}_{n} is divisible by 4141 if nn is divisible by kk
number theorydivisiblerepunit