3. 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 $\qquad$
Let the integer be A, when divided by $\frac{3}{5}, \frac{5}{7}, \frac{7}{9}, \frac{9}{11}$ respectively, the quotients obtained are $\frac{5}{3} A, \frac{7}{5} A, \frac{9}{7} A, \frac{11}{9} A$;
$$
\frac{5}{3} A=1+\frac{2}{3}+\frac{5}{3}(A-1), \frac{7}{5} A=1+\frac{2}{5}+\frac{7}{5}(A-1), \frac{9}{7} A=1+\frac{2}{7}+\frac{9}{7}(A-1), \frac{11}{9} A=1+\frac{2}{9}+\frac{11}{9}(A-1)
$$

Obviously, when $A-1$ is a multiple of $[3,5,7,9]$, it satisfies the condition. Therefore, $A-1=315, A=316$.