If A381 is divisible by 11, find the value of the smallest natural number A?

For the smallest value of A
Sum of digits at odd places - Sum of digits at even places = 0 or 11
A + 8 - (3 + 1) = 0 or 11
A + 4 = 0 gives A = - 4 not acceptable
A + 4 = 11 gives A = 7
Therefore A = 7 is the smallest natural number
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If the difference of the sum of the digits in the odd places and sum of the digits in even place is zero or divisible by 11, a number is divisible by 11.
So, (A + 8) - (3 + 1) = 0 or multiple of 11.
To get the difference 0 or multiple of 11, we require 7 at the place of A.
So, sum of odd place - sum of even place
= 15 - 4 = 11. And this is divisible by 11.