2. Determine the smallest natural number that is divisible by 13, and when divided by the numbers $2,3, \ldots, 11,12$ leaves a remainder of 1.
2. If $x$ is the sought number, then from the condition of the problem it follows that $x-1$ is divisible by $2,3, \ldots, 12$, hence

$$
27720=\mathrm{LCM}(2,3, \ldots, 12) \mid x-1
$$

In other words, $x=27720 k+1$ for some natural number $k$. By substituting $k=1,2,3 \ldots$ until we find the first case where $x$ is divisible by 13, we see that this happens for $k=3$, and then we have $x=27720 \cdot 3+1=83161$.