9. (40 points) Given the following numbers: 20172017, 20172018, 20172019, 20172020, and 20172021. Is there a number among them that is coprime with all the others? If so, which one?
Answer: $20172017, 20172019$.

Solution: The given numbers are five consecutive natural numbers. The even numbers 20172018 and 20172020 do not fit.

Adjacent numbers are always coprime:

$$
n=m q, \quad n+1=m p, \quad 1=m(p-q) \Rightarrow m=1
$$

Adjacent odd numbers are also always coprime:

$$
2 k+1=m q, \quad 2 k+3=m p, \quad 2=m(p-q), \quad m \neq 2 \Rightarrow m=1
$$

Therefore, the number 20172019 is guaranteed to be coprime with the others.

It remains to check the numbers 20172017 and 20172021. The number 20172021, like the number 20172018, is divisible by 3, so it does not fit. However, $20172017=2017 \cdot 137 \cdot 7$ is also coprime with the others.