1. There are 2008 viewing stations set up along a circular marathon track, labeled as $c_{1}, c_{2}, \cdots, c_{2008}$ in a clockwise direction. These stations divide the track into 2008 segments. An athlete places mascots numbered $1, 2, \cdots, 2008$ at these stations in the following manner: He first places the 1st mascot at $c_{1}$, then runs clockwise over 29 segments and places the 2nd mascot at the station $c_{30}$; he then runs clockwise over 29 segments again and places the 3rd mascot at the station $c_{59}$, and so on. What is the number of the mascot placed at station $c_{2008}$? 
(A) 2008 (B) 1896 (C) 1732 (D) 1731
- 1. C.

Let the mascot number placed at site $c_{k}$ be $x$, $x \in\{1,2, \cdots, 2008\}$. Then $29(x-1)+1 \equiv k(\bmod 2008)$.
When $k=2008$, we get
$29(x-1)+1 \equiv 0(\bmod 2008)$.
Let $29(x-1)+1=2008 y$, i.e., $x=\frac{2008 y-1}{29}+1=69 y+1+\frac{7 y-1}{29}$.
It is easy to see that when $y=-4+29 t, t \in \mathbf{Z}$, $x \in \mathbf{Z}$.
Taking $t=1$, we get $y=25$, at this time,
$$
x=69 y+1+\frac{7 y-1}{29}=1732 .
$$