document.write( "Question 1050569: What is the least possible difference between the sum
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Algebra.Com's Answer #666193 by Edwin McCravy(20060)\"\" \"About 
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document.write( "It would seem that it would be 6 because \r\n" );
document.write( "(5+7)-(2+4) = 12-6 = 6\r\n" );
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document.write( "Let's prove that:\r\n" );
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document.write( "Let the two consecutive odd integers be 2x+1 and 2x+3,\r\n" );
document.write( "Let the two consecutive even integers be 2y and 2y+2,\r\n" );
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document.write( "where x and y are integers.\r\n" );
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document.write( "Since both odd integers are greater than both even integers,\r\n" );
document.write( "the smaller odd integer must be greater than the larger \r\n" );
document.write( "even integer, so\r\n" );
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document.write( "2x+1 > 2y+2\r\n" );
document.write( "2x-2y > 1\r\n" );
document.write( "x-y > 0.5\r\n" );
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document.write( "so the least x-y can be is 1.\r\n" );
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document.write( "It CAN be 1 because in the example mentioned at the top,\r\n" );
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document.write( "(5+7)-(2+4) = 12-6 = 6\r\n" );
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document.write( "5=2*2+1, 7=2*2+3, 2=2*1, 4=2*2, the case when x=2 and y=1,\r\n" );
document.write( "and thus x-y = 2-1 = 1.\r\n" );
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document.write( "The difference between the sum of those 2 consecutive odd \r\n" );
document.write( "integers and the sum of the 2 consecutive even integers is\r\n" );
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document.write( "(2x+1 + 2x+3) - (2y + 2y+2) = (4x+4) - (4y+2) =\r\n" );
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document.write( "4x+4 - 4y-2 = 4x-4y+2 = 4(x-y)+2,\r\n" );
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document.write( "The least x-y can be is 1, and it CAN be 1, so\r\n" );
document.write( "the least their difference 4(x-y)+2 can be is when x-y=1,\r\n" );
document.write( "or 4(1)+2 = 6\r\n" );
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document.write( "So we have proved that the least possible difference between \r\n" );
document.write( "the sum of 2 consecutive odd integers and the sum of 2 \r\n" );
document.write( "consecutive even integers when both odd integers are greater \r\n" );
document.write( "than both even integers is 6.\r\n" );
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document.write( "Edwin
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