26.2. Find the smallest natural number divisible by 11, which after being increased by 1, would be divisible by 13.

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\text { (7-8 grades) }
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26.2. Let the desired number be denoted by $n$. According to the problem, we have: $n=11 k$ and $n+1=13 m$, where $k$ and $m$ are natural numbers. Then $11 k=13 m-1$ or $(13-2) k=13 m-1$, from which we get $13(k-m)=2 k-1$. In the obtained equation, the left side is divisible by 13. Therefore, the right side must also be divisible by 13. The smallest value of $k$ for which this is true is 7. Then $n=11 \cdot 7=77$.