Find the smallest four-digit number, all digits of which are different and not equal to zero

Find the smallest four-digit number, all digits of which are different and not equal to zero, such that the sum of all its digits is divisible by each of them.

According to the problem statement, a four-digit number consists of different digits and there is no 0 among these digits.

For this number to be the smallest, it must start with the smallest possible digits.

In the first place we will put the smallest number 1, then number 2, since 0 cannot be, and then number 3.

The required number will look like 123 *.

The sum of the digits 1 + 2 + 3 = 6 is divisible by 1, 2 and 3.

Instead of *, select the smallest of the remaining digits so that the sum 1 + 2 + 3 + * is divisible by this digit.

Take the number 4: 1 + 2 + 3 + 4 = 10 is not divisible by 3 and 4.

Take the number 5: 1 + 2 + 3 + 5 = 11 is divisible only by 1.

Take the number 6: 1 + 2 + 3 + 6 = 12 is divisible by 1, 2, 3, and 6.

Answer: 1236.