document.write( "Question 1151145: There are two triples of positive integers (a,b,c and d,e,f) such that a^2+b^2+c^2 = 86 and d^2+e^2+f^2 = 86. Numbers should not repeat within each triplet. Evaluate the expression | abc-def | \n" ); document.write( "

Algebra.Com's Answer #772784 by Edwin McCravy(20060) $\"\"$ $\"About$

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There are two triples of positive integers (a,b,c and d,e,f) such that
\n" ); document.write( "a^2+b^2+c^2 = 86 and d^2+e^2+f^2 = 86. Numbers should not repeat within each
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document.write( "We'll assume a < b < c and d < e < f\r\n" );
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document.write( "a,b,c,d,e,f all have to be 9 or less.  The squares are\r\n" );
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document.write( "1,4,9,16,25,36,49,64,81 \r\n" );
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document.write( "Three of these must have sum 86\r\n" );
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document.write( "We try c = 9\r\n" );
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document.write( "a² + b² + c² = 86\r\n" );
document.write( "a² + b² + 9² = 86\r\n" );
document.write( "a² + b² + 81 = 86\r\n" );
document.write( "a² + b² = 5\r\n" );
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document.write( "We see that the first two, 1 and 4, have sum 5\r\n" );
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document.write( "so (a,b,c) or (d,e,f) can be (1,4,9)\r\n" );
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document.write( "We try c = 8\r\n" );
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document.write( "a² + b² + c² = 86\r\n" );
document.write( "a² + b² + 8² = 86\r\n" );
document.write( "a² + b² + 64 = 86\r\n" );
document.write( "a² + b² = 22\r\n" );
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document.write( "No two of these 1,4,9,16,25,36,49,64,81 add to 22, so\r\n" );
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document.write( "we try c = 7\r\n" );
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document.write( "a² + b² + c² = 86\r\n" );
document.write( "a² + b² + 7² = 86\r\n" );
document.write( "a² + b² + 49 = 86\r\n" );
document.write( "a² + b² = 37\r\n" );
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document.write( "We see that 1+36=37, \r\n" );
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document.write( "so (a,b,c) or (d,e,f) can be (1,6,7)\r\n" );
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document.write( "Let's let (a,b,c) = (1,4,9) and (d,e,f) = (1,6,7) \r\n" );
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document.write( "You calculate | abc-def |\r\n" );
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document.write( "Edwin

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