(related to Problem: The Mystic Eleven)
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 naught a digit) that is divisible by $11$ is $102,347,586,$ and the highest number possible, $987,652,413.$
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