To find the number of birdhouses, divide the length of the board by the length of each birdhouse: $10\frac{2}{5} \div 1\frac{3}{10}=\frac{52}{5} \div \frac{13}{10}=\frac{52}{5} \cdot \frac{10}{13}= \boxed{\frac{104}{13}=\frac{8\text{ birdhouses}}{1}}$.
