$\underline{\text { Folklore }}$

Find the number of solutions in natural numbers of the equation $\left[{ }^{x} / 10\right]=\left[{ }^{x} / 11\right]+1$.
Let $x=11 n+r$, where $n \geq 0,0 \leq r \leq 10$. Then $\left[{ }^{x} / 11\right]=n, n+1=\left[{ }^{x} / 10\right]=n+\left[{ }^{n+r} / 10\right]$, that is, $10 \leq n+r<20,10$ $-r \leq n \leq 19-r$. For each $r$ from 0 to 10, we get 10 solutions.

## Answer

110 solutions.