jmc

We know that the five-digit number must be divisible by each of its non-zero digits. We should include the digit zero since it does not have to be included as a possible divisor and it will keep the integer smaller. Knowing this, the least possible five-digit number that we could try is $10,234$. Any number we choose will be divisible by one. We also see that it is even, and therefore, divisible by two. However, the two-digit number formed by its last two digit ($34$) is not divisible by four, and therefore, neither is the five-digit number. We also see that the sum of the five digits is $10$, and since $10$ is not divisible by three, neither is the five-digit number. However, notice that by increasing the five-digit number by two to form the number $10,236$, we create another even number and increase the digit-sum to $12$ (which takes care of the number being divisible by three). We have now eliminated the digit of four and added the digit of six, which is fine because $10,236$ being divisible by both two and three means that it is divisible by six. Our five-digit number is $\boxed{10,236}$.