Whats the smallest number ("the magic number") that can satisfy these conditions?

- its positive

- it is not divisible by any number below 20 (excluding 1)

- the average of the digits is an even number between 3 and 9

All even numbers between 20 and 29 are excluded, for they are (obviously!) divisible by 2, and the average of the digits of the odd numbers between 20 and 29 isn't even, so we should start looking at 30. The first number greater than 30 that satisfies all conditions is 37, so that's the answer.