Tokarev S.i.

The set of five-digit numbers $\{N_1, \dots, N_k\}$ is such that any five-digit number, all of whose digits are in increasing order, coincides in at least one digit with at least one of the numbers $N_1, \dots, N_k$. Find the smallest possible value of $k$.
Let's prove that the condition is satisfied by the set consisting of a single number $13579$. Indeed, let

$\overline{a_1a_2a_3a_4a_5}$ be a five-digit number, the digits of which satisfy the inequalities $a_1 < a_2 < a_3 < a_4 < a_5$. Then, if $a_1 \ne 1$, we have $2 \leq a_1 < a_2$. If at the same time $a_2 \ne 3$, then $4 \le a_2 < a_3$.

![](https://cdn.mathpix.com/cropped/2024_05_06_388b3f8bbd0a17840215g-02.jpg?height=57&width=2004&top_left_y=1921&top_left_x=10)
i.e., the equality $a_5 = 9$, meaning the digits in the units place coincide.

## Answer

$\$ \mathrm{k}=1 \$$.
