document.write( "Question 828701: The sides of a triangle all have integral measure and can be represented by the expressions x + 10, 2x -3, and 4x. For how many values of x will the sum of the two longer sides be divisible by the shortest side? \n" ); document.write( "
Algebra.Com's Answer #499538 by Edwin McCravy(20063) $\"\"$ $\"About$
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document.write( "Since the sides must be positive integers, \r\n" );
document.write( "4x will always be greater than 2x-3.\r\n" );
document.write( "So we have three cases to consider:\r\n" );
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document.write( "Case 1:  \r\n" );
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document.write( "Since these are the sides of a triangle, the sum of the\r\n" );
document.write( "two shorter sides must by greater than the longest side\r\n" );
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document.write( "So there are no solutions in case 1, since those \r\n" );
document.write( "inequalities contradict each other.         \r\n" );
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document.write( "Case 2:  \r\n" );
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document.write( "          and \r\n" );
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document.write( "Since these are the sides of a triangle, the sum of the\r\n" );
document.write( "two smaller sides must be greater than the largest side\r\n" );
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document.write( "So we have the possibility \r\n" );
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document.write( "For x=4  ,,\r\n" );
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document.write( "2x-3=2(4)-3=8-3=5\r\n" );
document.write( "x+10=4+10=14\r\n" );
document.write( "4x=4(4)=16\r\n" );
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document.write( "The sides are 5,14,16   \r\n" );
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document.write( "The sum of the two longer sides is 30, which \r\n" );
document.write( "is divisible by the shortest side, 5.\r\n" );
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document.write( "So one solution is x=4 \r\n" );
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document.write( "For x=5  ,,\r\n" );
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document.write( "2x-3=2(5)-3=10-3=7\r\n" );
document.write( "x+10=5+10=15\r\n" );
document.write( "4x=4(5)=20\r\n" );
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document.write( "The sides are 7,15,20\r\n" );
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document.write( "The sum of the two longer sides is 35, which \r\n" );
document.write( "is divisible by the shortest side, 7.\r\n" );
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document.write( "So a second solution is x=5 \r\n" );
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document.write( "For x=6  ,,\r\n" );
document.write( "          9,16,24\r\n" );
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document.write( "The sum of the two longer sides is 40, which \r\n" );
document.write( "is NOT divisible by the shortest side, 9. So x=6\r\n" );
document.write( "is NOT a solution.       \r\n" );
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document.write( "So from case 1, we have exactly two solutions, x=4 and x=5\r\n" );
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document.write( "Case 3:  \r\n" );
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document.write( "Since these are the sides of a triangle, the sum of the\r\n" );
document.write( "two shorter sides must be greater than the longest side\r\n" );
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document.write( "So we have one possibility x=3\r\n" );
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document.write( ",,\r\n" );
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document.write( "2x-3=2(3)-3=6-3=3\r\n" );
document.write( "4x=4(3)=12\r\n" );
document.write( "x+10=3+10=13\r\n" );
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document.write( "The sides are 3,12,13\r\n" );
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document.write( "The sum of the two longer sides is 25, which \r\n" );
document.write( "is NOT divisible by the shortest side, 3. So x=3\r\n" );
document.write( "is NOT a solution. \r\n" );
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document.write( "This exhausts all the possibilities, so the answer is \r\n" );
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document.write( "There are exactly two possibilities:\r\n" );
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document.write( "x=4 which yields sides 5, 14, 16\r\n" );
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document.write( "x=5 which yields sides 7, 15, 20 \r\n" );
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document.write( "Edwin
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