15. (6 points) The number of simplest proper fractions where the sum of the numerator and denominator is 2013 is. $\qquad$
15. (6 points) There are 600 simplest proper fractions whose sum of the numerator and denominator is 2013.
【Solution】Solution: The proper fractions whose sum of the numerator and denominator is 2013 are $\frac{1}{2012}, \frac{2}{2011}, \frac{3}{2010} \cdots \frac{1006}{1007}$, a total of 1006 fractions. $2013=3 \times 11 \times 61$, as long as the numerator is a multiple of the prime factors of 2013, the fraction is not in its simplest form. Since the sum of the numerator and denominator is 2013, if the numerator is a multiple of $3, 11, 61$, then the denominator must also be a multiple of 3, 11, or 61.
$$
\begin{array}{l}
{[1006 \div 3]=335, \quad[1006 \div 11]=91, \quad[1006 \div 61]=16,} \\
{[1006 \div 3 \div 11]=30, \quad[1006 \div 3 \div 61]=5, \quad[1006 \div 11 \div 61]=1,} \\
1006-335-91-16+30+5+1=600 .
\end{array}
$$

Therefore, the answer is: 600.