Algebra.Com's Answer #842892 by ikleyn(52878)![\"\"](\"/images/stars/stars-13.gif\")  
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document.write( "Find the difference  , where x and y are positive integers and  .
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document.write( "Your starting equation is \r\n" );
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document.write( "  .\r\n" );
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document.write( "Factor 14399 =  . Re-write equation in this equivalent form\r\n" );
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document.write( "  = .\r\n" );
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document.write( "Right side is divisible by 7, and one term in the left side is a multiple of 7;\r\n" );
document.write( "hence, the term  is divisible by 7.\r\n" );
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document.write( "So, I write y = 7z, where z is some positive integer number.\r\n" );
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document.write( " I can not write x = 7z, since then the degree of 7 will be too great on the left side.\r\n" );
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document.write( "Then the last equation takes the form\r\n" );
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document.write( "  = , or\r\n" );
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document.write( "  = .\r\n" );
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document.write( "Cancel factor 7 in both sides and get\r\n" );
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document.write( "  =  ,\r\n" );
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document.write( "  = .\r\n" );
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document.write( "Recall about the uniqueness of decomposition of integer numbers into the product of primes.\r\n" );
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document.write( "It implies that x= 11.\r\n" );
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document.write( "Then 7z^2 - x = 17; 7z^2 - 11 = 17; 7z^2 = 17 + 11 = 28; z^2 = 28/7 = 4; z=  = 2.\r\n" );
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document.write( "Thus x= 11; z= 2. Hence, y = 7z = 7*2 = 14.\r\n" );
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document.write( "The difference y-x is 14-11 = 3.\r\n" );
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document.write( "ANSWER. y-x = 3.\r\n" );
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document.write( "Solved.\r
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document.write( "I understand that I am helping somebody to solve a problem of a Math Olympiad level.\r
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