6. (20 points) Natural numbers $a, b$ satisfy $23a - 13b = 1$, find the minimum value of $a + b$.

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【Analysis】From $23 a-13 b=1$, we get $13 b=23 a-1=26 a-(3 a+1)$, leading to the form of the solution for $b$ in terms of $a$: $b=26 a \div 13-(3 a+1) \div 13=2 a-(3 a+1) \div 13$. Since $a$ and $b$ are both natural numbers, $3 a+1$ must be divisible by 13. Clearly, the smallest value for $a$ is $4$, and $b$ simultaneously achieves its minimum value of $b=7$, thus finding the minimum value of $a+b$.

【Solution】Solution: From $23 a-13 b=1$, we get $13 b=23 a-1=26 a-(3 a+1)$
leading to $b=26 a \div 13-(3 a+1) \div 13=2 a-(3 a+1) \div 13$
For $3 a+1$ to be divisible by 13,
clearly the smallest value for $a$ is $4$, and $b$ simultaneously achieves its minimum value of $b=7$
Therefore, the minimum value of $a+b$ is $4+7=11$.