[ CaseAnalysis ]  $[$ Divisibility Rules (etc.) $]$  Author: Fomin S.B.  A natural number $n$ is written in the decimal system. It is known that if a digit appears in this representation, then $n$ is divisible by this digit (0 does not appear in the representation). What is the maximum number of different digits that this representation can contain?
If the digit 5 is included in the representation of a number, then the number must end in 5. Therefore, it is odd and, consequently, contains only odd digits. Thus, it cannot have more than five digits. If 5 does not appear in the decimal representation of the number, then it can include all other 8 digits. For example: 1471963248. This number is divisible by $1,2,3,4,6,7,8,9$.

## [ Dirichlet's Principle (other). ] Problem 98031 Topics: [Proof by contradiction] $[$ Combinations and permutations $]$ Author: Folklore 10 friends sent each other holiday cards, with each sending five cards. Prove that there will be a pair who sent cards to each other.  Solution  Suppose this is not the case, that is, there is no pair of friends who sent cards to each other. Each friend sent cards to five others, so they could have received no more than four cards. Therefore, the total number of cards received is no more than 40, while 50 cards were sent ($5 \cdot 10=50$). This is a contradiction.