10. Arrange all positive integers that are coprime with 70 in ascending order. The 2017th term of this sequence is $\qquad$
10. 5881.

It is easy to know that the number of positive integers not exceeding 70 and coprime with 70 is
$$
35-(7+5)+1=24 \text {. }
$$

Let the sequence of all positive integers coprime with 70, arranged in ascending order, be $\left\{a_{n}\right\}$. Then
$$
\begin{array}{l}
a_{1}=1, a_{2}=3, a_{3}=9, \cdots, a_{24}=69 . \\
\text { Let } A=\left\{a_{i} \mid 1 \leqslant i \leqslant 24, i \in \mathbf{N}\right\} .
\end{array}
$$

Notice that, for any $k 、 r \in \mathbf{N}$, we have
$$
(70 k+r, 70)=(r, 70) \text {. }
$$

Therefore, for each fixed non-negative integer $k$, when $r$ takes all values in the set $A$, there are 24 positive integers of the form $70 k+r$ that are coprime with 70. Thus, we obtain the 24 terms of the sequence $\left\{a_{n}\right\}$.
$$
\begin{array}{l}
\text { Also, } 2017=24 \times 84+1 \text {, hence, } \\
a_{2017}=70 \times 84+a_{1}=5881 .
\end{array}
$$