document.write( "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.
\n" ); document.write( "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. \n" ); document.write( "

Algebra.Com's Answer #280371 by richard1234(7193) $\"\"$ $\"About$

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Let $\"f%28x%29+=+sum%28a%5Bi%5D%2Ax%5Ei%2C+i+=+0%2C+n%29\"$ be a polynomial and $\"f%2810%29+=+sum%28a%5Bi%5D%2A10%5Ei%2C+i+=+0%2C+n%29\"$ be a base-10 number that is divisible by 9. It follows that $\"f%281%29\"$ is the sum of the digits, or $\"sum%28a%5Bi%5D%2C+i+=+0%2C+n%29\"$ . Then,\r
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= 0 (modulo 9) since all powers of 10 are congruent to 1 modulo 9. Hence, $\"f%2810%29\"$ and $\"f%281%29\"$ have the same residue modulo 9, and we are done.\r
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\n" ); document.write( "\n" ); document.write( "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. \n" ); document.write( "

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