document.write( "Question 252017: Q.3:-Show that there is no integer a such that a^2-3a-19 is divisible by 289? \n" ); document.write( "
Algebra.Com's Answer #183934 by Edwin McCravy(20055)\"\" \"About 
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Q.3:-Show that there is no integer a such that a^2-3a-19 is divisible by 289?
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document.write( "Let \"%28a%5E2-3a-19%29%2F289=k\"\r\n" );
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document.write( "We need to show that k is not an integer.\r\n" );
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document.write( "Assume, for contradiction, that it is an integer.\r\n" );
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document.write( "\"a%5E2-3a-19=289k\"\r\n" );
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document.write( "\"a%5E2-3a-19=289k\"\r\n" );
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document.write( "\"a%5E2-3a-19-289k=0\"\r\n" );
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document.write( "\"a%5E2-3a%2B%28-19-289k%29=0\"\r\n" );
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document.write( "For \"Ax%5E2%2BBx%2BC=0\" to have a rational solution,\r\n" );
document.write( "the discriminant \"B%5E2-4AC\" must be a perfect \r\n" );
document.write( "square.\r\n" );
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document.write( "Here \"A=1\", \"B=-3\",  \"C=-19-288k\"\r\n" );
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document.write( "discriminant = \"+%28-3%29%5E2-4%28-19-289k%29=9%2B76%2B1156k=85%2B1156k=17%285%2B68k%29\"\r\n" );
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document.write( "In order for this discriminant, \"17%285%2B68k%29\", to be a \r\n" );
document.write( "perfect square, since it has one factor of 17, the other\r\n" );
document.write( "factor \"5%2B68k\" must also have a factor of 17.  So there\r\n" );
document.write( "must be an integer n such that\r\n" );
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document.write( "\"5%2B68k=17n\"\r\n" );
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document.write( "\"5=17n-68k\"\r\n" );
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document.write( "\"5=17%28n-4k%29\"\r\n" );
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document.write( "\"5%2F17=n-4k\"\r\n" );
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document.write( "But the right side is an integer but the left side is not.\r\n" );
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document.write( "This contradicts our assumption that the discriminant is a perfect\r\n" );
document.write( "square.   \r\n" );
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document.write( "Therefore \"a\" cannot be a rational number.  And since \"a\" is\r\n" );
document.write( "not a rational number, it certainly cannot be an integer.\r\n" );
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document.write( "Edwin
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