3. (3 points) Some integers, when divided by $\frac{5}{7}, \frac{7}{9}, \frac{9}{11}, \frac{11}{13}$ respectively, yield quotients that, when expressed as mixed numbers, have fractional parts of $\frac{2}{5}, \frac{2}{7}, \frac{2}{9}, \frac{2}{11}$ respectively. The smallest integer greater than 1 that satisfies these conditions is $\qquad$.
【Analysis】Let this number be $x$, then: $x \div \frac{5}{7}=\frac{7}{5} x, x \div \frac{7}{9}=\frac{9}{7} x, x \div \frac{9}{11}=\frac{11}{9} x, x \div \frac{11}{13}=\frac{13}{11} x$, therefore this number should be the least common multiple of the denominators +1, according to this to solve.
【Solution】Solution: Let this number be $x$, then:
$$
\begin{array}{l}
x \div \frac{5}{7}=\frac{7}{5} x, \\
x \div \frac{7}{9}=\frac{9}{7} x, \\
x \div \frac{9}{11}=\frac{11}{9} x \\
x \div \frac{11}{13}=\frac{13}{11} x
\end{array}
$$

Therefore, this number should be the least common multiple of the denominators +1,
that is $5 \times 7 \times 9 \times 11+1=3465+1=3466$
Answer: The smallest integer greater than 1 that meets the condition is 3466.
The answer is: 3466.