9. (10 points) Among the natural numbers that leave a remainder of 1 when divided by 7 and also leave a remainder of 1 when divided by 11, the smallest natural number greater than 1 is $\qquad$
【Analysis】From the conditions that the number leaves a remainder of 1 when divided by 7 and also leaves a remainder of 1 when divided by 11, we can conclude: This natural number greater than 1 is the least common multiple of 7 and 11 plus 1. Since 7 and 11 are coprime, their least common multiple is 77, and then adding 1.

【Solution】Solution: $7 \times 11+1=78$;
Answer: The smallest number is 78;
Therefore, the answer is: 78.