3. (10 points) Some integers, when divided by $\frac{3}{5}, \frac{5}{7}, \frac{7}{9}, \frac{9}{11}$ respectively, yield quotients that, when expressed as mixed numbers, have fractional parts of $\frac{2}{3}, \frac{2}{5}, \frac{2}{7}, \frac{2}{9}$ respectively. The smallest integer greater than 1 that satisfies these conditions is $\qquad$ .
【Analysis】Observe the fractional part of the obtained quotients: $\frac{2}{3}, \frac{2}{5}, \frac{2}{7}, \frac{2}{9}$, the numerators are all 2, and the denominators are the numerators of these fractions $\frac{3}{5}, \frac{5}{7}, \frac{7}{9}, \frac{9}{11}$. Therefore, we just need to find the least common multiple of $3, 5, 7, 9$ and add 1.
【Solution】The least common multiple of $3, 5, 7, 9$ is: $3 \times 3 \times 5 \times 7=315$, so the smallest integer greater than 1 that meets the condition is: $315+1=316$; hence the answer is: 316.