jmc

We examine only the factors of $90,91,92,93,$ and $94$ that are less than $13$, because for a year to be lucky, it must have at least one factor between $1$ and $12$ to represent the month. $90$ has factors of $1,2,3,5,6,9,$ and $10$. Dividing $90$ by the last number $10$, gives $9$. Thus, $10/9/90$ is a valid date, and $1990$ is a lucky year. $91$ has factors of $1$ and $7$. Dividing $91$ by $7$ gives $13$, and $7/13/91$ is a valid date. Thus, $1991$ is a lucky year. $92$ has factors of $1,2$ and $4$. Dividing $92$ by $4$ gives $23$, and $4/23/92$ is a valid date. Thus, $1992$ is a lucky year. $93$ has factors of $1$ and $3$. Dividing $93$ by $3$ gives $31$, and $3/31/93$ is a valid date. Thus, $1993$ is a lucky year. $94$ has factors of $1$ and $2$. Dividing $94$ by $2$ gives $47$, and no month has $47$ days. Thus, $1994$ is not a lucky year, and the answer is $\boxed{1994}$.