MathDB

A positive integer is called a repunit, if it is written only by ones. The repunit with

n

digits will be denoted as

\underbrace{{11\cdots1}}_{n}

. Prove that: α) the repunit

\underbrace{{11\cdots1}}_{n}

is divisible by

37

if and only if

n

is divisible by

3

b) there exists a positive integer

k

such that the repunit

\underbrace{{11\cdots1}}_{n}

is divisible by

41

n

is divisible by

k

Problem Statement