Most people know that if the sum of the digits in the odd places of any
number is the same as the sum of the digits in the even places, then the
number is divisible by 11 without remainder. Thus in 896743012 the odd
digits, 20468, add up 20, and the even digits, 1379, also add up 20.
Therefore the number may be divided by 11. But few seem to know that if
the difference between the sum of the odd and the even digits is 11, or
a multiple of 11, the rule equally applies. This law enables us to find,
with a very little trial, that the smallest number containing nine of
the ten digits (calling nought a digit) that is divisible by 11 is
102,347,586, and the highest number possible, 987,652,413.