document.write( "Question 1186694: How many integer solutions(a,b,c) make the equation a^2+b^2+c^2=121 true \n" ); document.write( "

Algebra.Com's Answer #817643 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "How many integer solutions(a,b,c) make the equation a^2+b^2+c^2=121 true
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document.write( "The basic solutions are these 3 triples:\r\n" );
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document.write( "a       b       c\r\n" );
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document.write( "0       0       11     (1)\r\n" );
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document.write( "2       6        9     (2)\r\n" );
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document.write( "6	6	 7     (3)\r\n" );
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document.write( "1)  Triple (1) creates  3 permutations, that are the solutions, too.\r\n" );
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document.write( "                               Playing with the signs, each of these permutations provides  2 triples, that are the solutions, too.\r\n" );
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document.write( "    So, triple (1) produces 2*3 = 6 triples that are the solutions.\r\n" );
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document.write( "2)  Triple (2) creates  6 permutations, that are the solutions, too.\r\n" );
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document.write( "                               Playing with the signs, we get  = 6*8 = 48 triples, that are the solutions, too.\r\n" );
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document.write( "    So, triple (2) produces 48 triples that are the solutions.\r\n" );
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document.write( "3)  Triple (3) creates  3 distinguishable permutations, that are the solutions, too.\r\n" );
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document.write( "                               Playing with the signs, each of these 3 distinguishable permutations provides 8 triples, that are the solutions, too.\r\n" );
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document.write( "    So, triple (3) produces  = 24 triples that are the solutions.\r\n" );
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document.write( "In all,  there are   6 + 48 + 24 = 78   different integer triples that are the solutions.      ANSWER\r\n" );
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