10. (10 points) Simplify each of the fractions $\frac{1}{2014}, \frac{2}{2014}, \cdots, \frac{2012}{2014}, \frac{2013}{2014}$ to their simplest form, and find the sum of all the fractions that still have 2014 as their denominator after simplification.
【Analysis】Since $2014=2 \times 1007=2 \times 19 \times 53$, we need to find the number of multiples of $2$, $19$, $53$, $2 \times 19$, $2 \times 53$, $19 \times 53$, and $2 \times 19 \times 53$, respectively, to determine the number of denominators that are not 2014, and thus solve the problem.
【Solution】Solution: $2014=2 \times 1007=2 \times 19 \times 53$,
Multiples of 2: 1007,
Multiples of 19: 106,
Multiples of 53: 38,
Multiples of $2 \times 19$: 53,
Multiples of $2 \times 53$: 19,
Multiples of $19 \times 53$: 2,
Multiples of $2 \times 19 \times 53$: 1,
The number of denominators that are not 2014 is:
$1007+106+38-53-19-2+1=1078$ (cases),
Excluding $\frac{2014}{2014}$, the total is: $1078-1=1077$ (cases).
The number of simplest fractions with a denominator of 2014 is:
$2013-1077=936$ (cases)
The sum is: $936 \div 2=468$.