Consider $\overline{abcde}$. We need $\overline{bcde} \mid 10^4a$. Since $9\cdot 2^4 < 1000$ we need $5\mid \overline{bcde}$, thus $2\nmid \overline{bcde}$. $a=1$ is too small, since $5^4<1000$. Thus $a=2,4,8$ are also inconvenient. $a=3$ is inconvenient, since $3\cdot 5^4 = 1875$, and $75\nmid 800$. Thus $a=6$ is also inconvenient. $a=5$ is convenient, since $5\cdot 5^4 = 3125$, and $\overline{53125}$ satisfies. $a=7$ is inconvenient, since $7\cdot 5^3 < 1000$, while $7\cdot 5^4 = 4375$, and $375\nmid 4000$. $a=9$ is convenient, since $9\cdot 5^3 = 1125$, and $\overline{91125}$ satisfies, while $9\cdot 5^4 = 5625$, and $\overline{95625}$ also satisfies.