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

Rem(123, 9) = 6

Likewise

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

Proof

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.

Conclusions