document.write( "Question 1037532: How many positive integers $\"+n+\"$ less than 2015 have the property that $\"+1%2F3+%2B+1%2Fn+\"$ can be simplified to a fraction with denominator less than $\"+n+\"$ ? \n" ); document.write( "

Algebra.Com's Answer #652184 by Edwin McCravy(20056) $\"\"$ $\"About$

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document.write( "Certainly the denominator 3n is greater than n.  Therefore\r\n" );
document.write( "the fraction must reduce.  Thus let d > 1 be the greatest\r\n" );
document.write( "common divisor of n+3 and 3n, the numerator and denominator.\r\n" );
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document.write( "Then there exist positive integers k,m such that\r\n" );
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document.write( "n+3 = kd and 3n = md\r\n" );
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document.write( "So the fraction  reduces to , where m < n.\r\n" );
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document.write( "Then n = kd-3, and by substitution\r\n" );
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document.write( "3(kd-3) = md\r\n" );
document.write( "3kd - 9 = md\r\n" );
document.write( " 3kd-md = 9\r\n" );
document.write( "d(3k-m) = 9\r\n" );
document.write( "      d = \r\n" );
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document.write( "So 3k-m is a divisor of 9, either 1, 3, or 9.\r\n" );
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document.write( "Since d > 1, 3k-m ≠ 9. So d is one of the\r\n" );
document.write( "other two possibilities, d=3 or d=9\r\n" );
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document.write( "If d=3, then 3k-m = 3\r\n" );
document.write( "   3k = m+3\r\n" );
document.write( "    k = \r\n" );
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document.write( "Since n+3 = kd,\r\n" );
document.write( "n+3 = *3\r\n" );
document.write( "n+3 = m+3\r\n" );
document.write( "  n = m\r\n" );
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document.write( "But that contradicts m < n\r\n" );
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document.write( "So d=9, and 3k-m = 1\r\n" );
document.write( "m = 3k-1\r\n" );
document.write( "3n = md = dm\r\n" );
document.write( "3n = 9(3k-1)\r\n" );
document.write( " n = 3(3k-1)\r\n" );
document.write( " n = 9k-3\r\n" );
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document.write( "Therefore n must be in the sequence 6,15,24,...,9k-3,...\r\n" );
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document.write( "n = 9k-3 ≤ 2015\r\n" );
document.write( "      9k ≤ 2018\r\n" );
document.write( "       k ≤ 224.2222...\r\n" );
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document.write( "So maximum value of k is 224,\r\n" );
document.write( "thus the answer is 224.\r\n" );
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document.write( "Edwin

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