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The Remainder upon Division by Nine of
Any Rearrangement of the Digits of any
Number is the Same as that of the Number


Let the number be 123. Its remainder upon division by 9 is 6. In symbols

Rem(123, 9) = 6


Rem(132, 9) = 6
Rem(213, 9) = 6
Rem(231, 9) = 6
Rem(312, 9) = 6
Rem(321, 9) = 6


The digit sum of a number is the repeated sum of the digits of a number; i.e., the sum of the digits is computed, then the sum of the digits of that sum is computed and this process is repeated until the result is a single digit. This digit sum is the same as the remainder for the division of the number by nine with the provision that a digit sum of nine is equivalent to a remainder of zero for division by nine.

The digit sum of any rearrangement of a number is the same as that of the number.

Consider the number 314159. Its remainder upon division by 9 is 5. It sum of its digits is 23 so its digit sum is 5. The remainder upon division of 341159 by 9 is also 5 as is the case for 341195 as well. nnnnnnnnnnnnnn

The proposition is also true for numbers to base 8 for division by 7. Likewise it is true for numbers to the bases of 7 and 6 for division by 6 and 5, respectively. For numbers to bases below 6 the situation is more complicated.


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