document.write( "Question 398690: Ten-digit number. Find a 10-digit number whose first digit
\n" ); document.write( "is the number of 1’s in the 10-digit number, whose second
\n" ); document.write( "digit is the number of 2’s in the 10-digit number, whose
\n" ); document.write( "third digit is the number of 3’s in the 10-digit number, and
\n" ); document.write( "so on. The ninth digit must be the number of nines in the
\n" ); document.write( "10-digit number and the tenth digit must be the number of
\n" ); document.write( "zeros in the 10-digit number \n" ); document.write( "

Algebra.Com's Answer #282726 by richard1234(7193) $\"\"$ $\"About$

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I'll denote the digits $\"a%5B1%5D\"$ , $\"a%5B2%5D\"$ , ..., $\"a%5B10%5D\"$ . Since there are $\"a%5B1%5D\"$ 1's, $\"a%5B2%5D\"$ 2's, ..., $\"a%5B10%5D\"$ 0's, then $\"sum%28a%5Bi%5D%2C+i+=+1%2C+10%29+=+10\"$ because there are ten digits in the number.\r
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\n" ); document.write( "\n" ); document.write( "Suppose $\"a%5B10%5D+=+0\"$ . Then a contradiction would result, since there is at least one zero, so $\"a%5B10%5D+%3E=+1\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Suppose $\"a%5B10%5D+=+1\"$ . Then, this implies that there is exactly one zero within the number, and all the other $\"a%5Bi%5D\"$ 's are at least one, which also creates a contradiction due to the sum of digits. This can also apply to higher values of $\"a%5B10%5D\"$ . If $\"a%5B10%5D+=+n\"$ , then $\"sum%28a%5Bi%5D%2C+i+=+1%2C+9%29+=+10-n\"$ . Also, if n of the numbers { $\"a%5B1%5D\"$ , $\"a%5B2%5D\"$ , ..., $\"a%5B9%5D\"$ } are zero, then 9-n of these numbers are greater than or equal to 1, and they sum up to 10-n.\r
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\n" ); document.write( "\n" ); document.write( "By the Pigeonhole principle, exactly one of the 9-n numbers is equal to 2 and all other nonzero numbers are equal to 1 (this means, 8-n numbers equal to 1). Therefore the number has n zeros, 8-n 1's and one 2. We can easily guess and check based on the value of n, which can only range between 2 and 8. We see that the number 2100010006 satisfies all the given constraints, which happens when $\"n+=+6\"$ . \n" ); document.write( "

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