document.write( "Question 339141: How many possible solution are there for:-
\n" ); document.write( "2x+3y+12z=180 ,where x,y, z are all positive integers???? \n" ); document.write( "
Algebra.Com's Answer #243168 by Edwin McCravy(20056)\"\" \"About 
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document.write( "2x + 3y + 12z = 180\r\n" );
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document.write( "2x = 180 - 3y - 12z\r\n" );
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document.write( "2x = 3(60 - y - 12z)\r\n" );
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document.write( "Let positive integer 60-y-12z = A then\r\n" );
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document.write( "So 2x = 3A\r\n" );
document.write( "   2x = 2A + A\r\n" );
document.write( "    x = A + A/2\r\n" );
document.write( "x - A = A/2\r\n" );
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document.write( "Since x - A is a positive integer, so is A/2, say B, and A = 2B\r\n" );
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document.write( "x - A = A/2\r\n" );
document.write( "x - 2B = 2B/2\r\n" );
document.write( "x - 2B = B\r\n" );
document.write( "     x = 3B\r\n" );
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document.write( "2x + 3y + 12z = 180\r\n" );
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document.write( "3y = 180 - 2x - 12z\r\n" );
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document.write( "3y = 2(90 - x - 6z)\r\n" );
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document.write( "Let positive integer 90-x-6z = C then\r\n" );
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document.write( "So 3y = 2C\r\n" );
document.write( "    y = 2C/3\r\n" );
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document.write( "Since y is a positive integer C must be divisible by 3, \r\n" );
document.write( "so C = 3D, so\r\n" );
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document.write( "   y = 2(3D)/3\r\n" );
document.write( "   y = 2D \r\n" );
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document.write( "So we substitute x = 3B and y = 2D into the original eqution:\r\n" );
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document.write( "2x + 3y + 12z = 180\r\n" );
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document.write( "2(3B) + 3(2D) + 12z = 180\r\n" );
document.write( "6B + 6D + 12z = 180\r\n" );
document.write( " B + D + 2z = 30\r\n" );
document.write( "         2z = 30 - (B + D)\r\n" );
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document.write( "The largest value 2z can take on is when B and D are both as small\r\n" );
document.write( "as can be, which is 1 each, so\r\n" );
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document.write( "         2z <= 30 - (1 + 1)\r\n" );
document.write( "         2z <= 28\r\n" );
document.write( "          z <= 14\r\n" );
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document.write( "So we know that z can take on any integer value from 1 through 14\r\n" );
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document.write( " B + D = 30 - 2z\r\n" );
document.write( "     B = 30 - 2z - D\r\n" );
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document.write( "For any of those 14 values of z, B can be chosen anywhere from 1\r\n" );
document.write( "through when D is the smallest value 1, or (30 - 2z - 1) or (29 - 2z) \r\n" );
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document.write( "Therefore the number of solutions is\r\n" );
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document.write( "\"sum%28%2829-2z%29%2C+z=1%2C+14+%29\", which equals\r\n" );
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document.write( "\"sum%2829%2C+z=1%2C+14+%29-sum%28%282z%29%2C+z=1%2C+14+%29\", which equals\r\n" );
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document.write( "\"14%2A29+-+2sum%28z%2Cz=1%2C14%29\"\r\n" );
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document.write( "The sum of the first n integers is \"S%5Bn%5D=%28n%28n%2B1%29%29%2F2\" so\r\n" );
document.write( "the sum of the first 14 integers is \"%2814%2A15%29%2F2+=+105\",\r\n" );
document.write( "so we can replace the summation by 105\r\n" );
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document.write( "\"14%2A29+-+2%2A105+=+196\"\r\n" );
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document.write( "So the number of positive integer solutions is 196.\r\n" );
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document.write( "Edwin
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