SOLUTION: Prove the well known result that the remainder on dividing a number by 9 is the same as the remainder on dividing the sum of its digits by 9. Show that the result may be generaliz

Question 394982: Prove the well known result that the remainder on dividing a number by 9 is the same as the remainder on dividing the sum of its digits by 9.
Show that the result may be generalized as follows; if a number is divided by s then the remainder is the same as the remainder on dividing bt s the sum of its digits, when it is expressed to the base s+1.
Answer by richard1234(7193) (Show Source): You can put this solution on YOUR website!
Let be a polynomial and be a base-10 number that is divisible by 9. It follows that is the sum of the digits, or . Then,

= 0 (modulo 9) since all powers of 10 are congruent to 1 modulo 9. Hence, and have the same residue modulo 9, and we are done.

To generalize to any base, simply replace 10 with base b+1. The same result should follow, since b+1 is always 1 modulo b.
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