big list - Interesting applications of the Pigeon-hole Principle

I think the solutions of these questions are very interesting (by using pigeon-hole principle), first question is easy, but second question is more advanced:

1) For any integer $n$, There are infinite integer numbers with digits only $0$ and $1$ where
they are divisible to $n$.

2) For any sequence $s=a_1a_2cdots a_n$, there is at least one $k$, such that $2^k$ begin with $s$.

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