Let $\overline{abcd}$ be such a number. Since $\overline{abcd}$ is divisible by $\overline{abc}$, we have $d = 0$. Since $\overline{abcd} = \overline{abc0}$ is divisible by $\overline{abd} = \overline{ab0}$, we have $c = 0$. Since $\overline{abcd} = \overline{ab00}$ is divisible by $\overline{acd} = \overline{a00}$ and by $\overline{bcd} = \overline{b00}$, the number $\overline{ab}$ is divisible by $a$ and $b$. So $b = ax$ and $10a = by$ with integer $x$ and $y$. Therefore $10a = axy$, whence $xy = 10$.

If $x = 1, y = 10$, then $a = b$, which gives 9 possible numbers: 1100, 2200, 3300, 4400, 5500, 6600, 7700, 8800, 9900.

If $x = 2, y = 5$, then $2a = b$, which gives 4 possibilities: 1200, 2400, 3600, 4800.

If $x = 5, y = 2$, then $5a = b$, which gives 1 number: 1500.

The case $x = 10, y = 1$ is impossible, since $a$ and $b$ must be one-digit numbers.