document.write( "Question 1006733: The number 2015 can be written as the sum of two or more consecutive
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Algebra.Com's Answer #623036 by Edwin McCravy(20056)\"\" \"About 
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document.write( "Let x represent the smallest integer in any case of consecutive\r\n" );
document.write( "integers with sum 2015.\r\n" );
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document.write( "The sum of n consecutive positive integers beginning with x, \r\n" );
document.write( "using the sum formula for an arithmetic sequence \r\n" );
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document.write( "\"s%5Bn%5D=expr%28n%2F2%29%282a%5B1%5D%2B%28n-1%29d%5E%22%22%29\"\r\n" );
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document.write( "with a1=x and d=1\r\n" );
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document.write( "\"%28n%2F2%29%282x%2B%28n-1%291%29\"\r\n" );
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document.write( "Setting that equal to 2015\r\n" );
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document.write( "\"%28n%2F2%29%282x%2B%28n-1%29%5E%22%22%29\"\"%22%22=%22%22\"\"2015\"\r\n" );
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document.write( "Multiplying both sides by 2 to clear the fraction\r\n" );
document.write( "and simplifying:\r\n" );
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document.write( "\"n%282x%2Bn-1%29\"\"%22%22=%22%22\"\"4030\"\r\n" );
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document.write( "\"n%282x%2Bn-1%29\"\"%22%22=%22%22\"\"4030\"\r\n" );
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document.write( "\"2nx%2Bn%5E2-n%29\"\"%22%22=%22%22\"\"4030\"\r\n" );
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document.write( "\"2nx\"\"%22%22=%22%22\"\"4030%2Bn-n%5E2\"\r\n" );
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document.write( "\"x+=+%284030%2Bn-n%5E2%29%2F%282n%29\", the smallest integer.\r\n" );
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document.write( "From \"n%282x%2Bn-1%29\"\"%22%22=%22%22\"\"4030\",\r\n" );
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document.write( "we see that n, the number of terms, and 2x+n-1 make up a factor \r\n" );
document.write( "pair of 4030.  \r\n" );
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document.write( "We can factor 4030 into a pair of factors the following 8 ways:\r\n" );
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document.write( "n×(2x+n-1) = 4030, x = (4030+n-n^2)/(2n) = smallest integer\r\n" );
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document.write( "1×4030             x = 2015  <--ignore since only 1 integer, we need 2 or more  \r\n" );
document.write( "2×2015             x = 1007                 \r\n" );
document.write( "5×806              x =  401\r\n" );
document.write( "10×403             x =  197\r\n" );
document.write( "13×310             x =  149\r\n" );
document.write( "26×155             x =   65\r\n" );
document.write( "31×130             x =   50  \r\n" );
document.write( "62×65              x =    2\r\n" );
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document.write( "               total = 1871  \r\n" );
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document.write( "The total of the smallest integers x is 1871\r\n" );
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document.write( "Answer: 1841\r\n" );
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document.write( "Edwin
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