document.write( "Question 506992: Find all pairs of positive integers whose squares differ by 400. \n" ); document.write( "
Algebra.Com's Answer #340301 by Edwin McCravy(20054) $\"\"$ $\"About$
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document.write( "Let x and y be integers x > y, such that x² - y² = 400\r\n" );
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document.write( "       x² - y² = 400\r\n" );
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document.write( "(x - y)(x + y) = 400\r\n" );
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document.write( "(x - y) represents the smaller factor of 400 and (x + y) represents \r\n" );
document.write( "the larger factor.\r\n" );
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document.write( "400 can be the product of two integers only in these ways:\r\n" );
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document.write( "(x - y)(x + y) = 400 = 1*400, where (x - y) = 1 and (x + y) = 400\r\n" );
document.write( "(x - y)(x + y) = 400 = 2*200, where (x - y) = 2 and (x + y) = 200\r\n" );
document.write( "(x - y)(x + y) = 400 = 4*100, where (x - y) = 4 and (x + y) = 100\r\n" );
document.write( "(x - y)(x + y) = 400 = 5*80, where (x - y) = 5 and (x + y) = 80\r\n" );
document.write( "(x - y)(x + y) = 400 = 8*50, where (x - y) = 8 and (x + y) = 50\r\n" );
document.write( "(x - y)(x + y) = 400 = 10*40, where (x - y) = 10 and (x + y) = 40\r\n" );
document.write( "(x - y)(x + y) = 400 = 16*25, where (x - y) = 16 and (x + y) = 25\r\n" );
document.write( "(x - y)(x + y) = 400 = 20*20, where (x - y) = 20 and (x + y) = 20\r\n" );
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document.write( "We eliminate some of these:\r\n" );
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document.write( "If x + y is even, then so is x - y\r\n" );
document.write( "If x + y is odd, then so is x - y\r\n" );
document.write( "The product of two even numbers is even and\r\n" );
document.write( "the product of two odd numbers is odd, so since\r\n" );
document.write( "400 is even, we can rule out the three cases where \r\n" );
document.write( "an odd number appears as either factor. We can also rule \r\n" );
document.write( "out the case where both factors = 20, since that would \r\n" );
document.write( "give the pair 20 and 0, but 0 is not a positive integer.\r\n" );
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document.write( "so we have only thse 4 possible cases:\r\n" );
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document.write( "1.  (x - y)(x + y) = 400 = 2*200, where (x - y) = 2 and (x + y) = 200\r\n" );
document.write( "2.  (x - y)(x + y) = 400 = 4*100, where (x - y) = 4 and (x + y) = 100\r\n" );
document.write( "3.  (x - y)(x + y) = 400 = 8*50, where (x - y) = 8 and (x + y) = 50\r\n" );
document.write( "4.  (x - y)(x + y) = 400 = 10*40, where (x - y) = 10 and (x + y) = 40\r\n" );
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document.write( "In case 1, we have this system of equations:\r\n" );
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document.write( " with solution x = 101, y = 99 \r\n" );
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document.write( "In case 2, we have this system of equations:\r\n" );
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document.write( " with solution x = 52, y = 48\r\n" );
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document.write( "In case 3, we have this system of equations:\r\n" );
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document.write( " with solution x = 29, y = 21\r\n" );
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document.write( "In case 4, we have this system of equations:\r\n" );
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document.write( " with solution x = 25, y = 15\r\n" );
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document.write( "Edwin
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