jmc

To begin with, note that all the numbers in question have a 1 in the hundreds place, and every number is divisible by 1, so we do not need to check it. So we need to see under what circumstances the number is divisible by its tens and units digits.

Let the three-digit number be $\overline{1TU}.$ We can then divide into cases, based on the digit $T.$

Case 1: $T=0$.

We are looking for three-digit numbers of the form $\overline{10U}$ that are divisible by $U,$ or where $U=0.$ If $\overline{10U}$ is divisible by $U,$ then 100 is divisible by $U.$ Thus, the possible values of $U$ are 0, 1, 2, 4, and 5.

Case 2: $T=1$.

We are looking for three-digit numbers of the form $\overline{11U}$ that are divisible by $U,$ or where $U=0.$ If $\overline{11U}$ is divisible by $U,$ then 110 is divisible by $U.$ Thus, the possible values of $U$ are 0, 1, 2, and 5.

Case 3: $T=2$.

We are looking for three-digit numbers of the form $\overline{12U}$ that are divisible by $U,$ or where $U=0.$ If $\overline{12U}$ is divisible by $U,$ then 120 is divisible by $U.$ Also, $\overline{12U}$ must be divisible by 2, which means $U$ is even. Thus, the possible values of $U$ are 0, 2, 4, 6, and 8.

Case 4: $T=3$.

We are looking for three-digit numbers of the form $\overline{13U}$ that are divisible by $U,$ or where $U=0.$ If $\overline{13U}$ is divisible by $U,$ then 130 is divisible by $U.$ Also, $\overline{13U}$ must be divisible by 3. Thus, the possible values of $U$ are 2 and 5.

Case 5: $T=4$.

We are looking for three-digit numbers of the form $\overline{14U}$ that are divisible by $U,$ or where $U=0.$ If $\overline{14U}$ is divisible by $U,$ then 140 is divisible by $U.$ Also, $\overline{14U}$ must be divisible by 4. Thus, the possible values of $U$ are 0 and 4.

Case 6: $T=5$.

Since the three-digit number must be between 100 and 150, the only number in this case is 150.

Adding up the possibilities gives us $\boxed{19}$ possible three-digit numbers.

$\begin{array}{ccccccccc}100& 101& 102& & 104& 105& & & \\ 110& 111& 112& & & 115& & & \\ 120& & 122& & 124& & 126& & 128\\ & & 132& & & 135& & & \\ 140& & & & 144& & & & \\ 150& & & & & & & & \end{array}$