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2015 JBMO Shortlist, NT2
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 $41$ if $n$ is divisible by $k$
repunit divisible number theory modular arithmetic