jmc

$10101$ is clearly not divisible by $2$ or $5$. The sum of $10101$'s digits is $3$, so it is divisible by $3$, but not by $9$. $10101=3\cdot 3367$. $3367/7=481$ and $481/7=68\frac{5}{7}$ so $10101=3\cdot 7\cdot 481$ and $481$ is not divisible by any prime number less than $11$. Applying the divisibility test for 11, we have $4-8+1=-3$, which is not divisible by 11, so $481$ is not divisible by $11$ either. $481/13=37$ and $37$ is prime, so the prime factorization of $10101$ is $10101=3\cdot 7\cdot 13\cdot 37$. So, the sum of its smallest and largest prime factors is $3+37=\boxed{40}$.