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

(In reply to

some thoughts on the puzzle by K Sengupta)

he average of the digits is an even number between 3 and 9, so the possibilities are:

Average = 4, Possibilities = 17, 71, 35, 53, 44, 26, 62. Only 71 and 53 conforms to the given conditions, and the smallest is 53.

Average = 6, Possibilities = 39, 93,48, 84, 57, 75,66. Each of these are divisible by a number less than 20. This is a contradiction.

Average = 8. Possibilities = 79, 97, 88. Only 79 and 97 are valid of which 79 is the smallest.

Thus, of the above cases, we have 53 and 79 as the smallest numbers, the minimum amongst which is 53.

Consequently, the required number is 53.