document.write( "Question 1047136: I am thinking of 4 numbers. They are all different, they add up to 20 in total, and the sum of any 3 of the numbers is exactly divisible by the other number. What are the numbers I am thinking of? Please include how you solved it not just the answer! :) thanks! \n" ); document.write( "
Algebra.Com's Answer #662648 by Edwin McCravy(20059)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "Let the positive integers be A, B, C, and D.  \r\n" );
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document.write( "A + B + C + D = 20\r\n" );
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document.write( "Then there exist positive integers p,q,r and s,\r\n" );
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document.write( "A + B + C = pD\r\n" );
document.write( "A + B + D = qC\r\n" );
document.write( "A + C + D = rB\r\n" );
document.write( "B + C + D = sA\r\n" );
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document.write( "Then we have this system of 5 equations in 8 unknowns:\r\n" );
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document.write( "  A  +  B +  C +  D = 20\r\n" );
document.write( "  A  +  B +  C - pD =  0\r\n" );
document.write( "  A  +  B - qC +  D =  0\r\n" );
document.write( "  A  - rB +  C +  D =  0\r\n" );
document.write( "-sA  +  B +  C +  D =  0 \r\n" );
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document.write( "Subtracting the 2nd equation from the 1st equation:\r\n" );
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document.write( "D + pD = 20\r\n" );
document.write( "D(1+p) = 20, so D is a factor of 20\r\n" );
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document.write( "Subtracting the 3rd equation from the 1st equation:\r\n" );
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document.write( "C + qC = 20\r\n" );
document.write( "C(1+q) = 20, so C is a factor of 20\r\n" );
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document.write( "Subtracting the 4th equation from the 1st equation:\r\n" );
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document.write( "B + rB = 20\r\n" );
document.write( "B(1+r) = 20, so B is a factor of 20\r\n" );
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document.write( "Subtracting the 5th equation from the 1st equation:\r\n" );
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document.write( "A + sA = 20\r\n" );
document.write( "A(1+s) = 20, so A is a factor of 20\r\n" );
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document.write( "So all 4 numbers must be factors of 20.\r\n" );
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document.write( "The only factors or 20 are 1,2,4,5,10, and 20.\r\n" );
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document.write( "Studying those for a minute, we see that the only way \r\n" );
document.write( "to pick out 4 of them, all different, that have sum \r\n" );
document.write( "20, is to leave out the 2 and the 20, and have \r\n" );
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document.write( "1+4+5+10 = 20.\r\n" );
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document.write( "So the answer is {1,4,5,10}\r\n" );
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document.write( "Checking:\r\n" );
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document.write( "1+4+5 = 10 which is a multiple of 10.\r\n" );
document.write( "1+4+10 = 15 which is a multiple of 5.\r\n" );
document.write( "1+5+10 = 16 which is a multiple of 4.\r\n" );
document.write( "4+5+10 = 19 which is a multiple of 1.\r\n" );
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document.write( "There would be two other answers, if we didn't have to\r\n" );
document.write( "use all different numbers.  They could all be 5, since\r\n" );
document.write( "5+5+5+5=20. Also 2+4+4+10=20. \r\n" );
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document.write( "But since they must all be different, those won't do,\r\n" );
document.write( "and the only answer is {1,4,5,10} \r\n" );
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document.write( "Edwin
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