document.write( "Question 1088473: a*b=62217, c*d=52983, b*c=56637, d*a=58203, b*d=72819, c*a=45269 solve this a+b+c+d=? , A*B*C*D=? , A=?, B=?, C=?,D=? \n" ); document.write( "

Algebra.Com's Answer #702775 by ikleyn(52790) $\"\"$ $\"About$

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\n" ); document.write( "a*b=62217, c*d=52983, b*c=56637, d*a=58203, b*d=72819, c*a=45269 solve this a+b+c+d=? , A*B*C*D=? , A=?, B=?, C=?,D=?
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\n" ); document.write( "\n" ); document.write( "I will show you more simple and more straightforward solution.\r
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document.write( "0.  First of all,  EITHER  all four numbers a, b, c and d are POSITIVE, OR all four of them are NEGATIVE.\r\n" );
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document.write( "    Indeed, had two numbers a and b, for example, have different signs, their product ab would be negative.\r\n" );
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document.write( "    Since all given values of products are positive, the statement made is true.\r\n" );
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document.write( "    It means that there are two solutions:\r\n" );
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document.write( "         one solution with all four numbers positive, and\r\n" );
document.write( "         the second solution with all four numbers negative (opposite).\r\n" );
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document.write( "    For simplicity, let us start finding positive solution, and \r\n" );
document.write( "        then will take the opposite numbers as the second solution, for completeness.\r\n" );
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document.write( "1.  The most easy question is about the product a*b*c*d. \r\n" );
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document.write( "    It is equal  a*b*c*d = (a*b)*(c*d) = 62217*52983 = 3296443311.   (1)\r\n" );
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document.write( "2.  Now I want to find the value of \"a\".\r\n" );
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document.write( "    For it, I will collect all given products that DO NOT CONTAIN the multiplier (the factor) \"a\".\r\n" );
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document.write( "    b*c = 56637,    (2)\r\n" );
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document.write( "    c*d = 52983,    (3)\r\n" );
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document.write( "    b*d = 72819.    (4)\r\n" );
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document.write( "Now multiply  all three equations (2), (3) and (4) (both sides).  You will get\r\n" );
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document.write( "     = 56637*52983*72819,   or\r\n" );
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document.write( "     = 218515122014049,    which implies (after taking square root of both sides)\r\n" );
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document.write( "    b*c*d = 14782257     (since we are looking now for positive numbers only).\r\n" );
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document.write( "    Then a =  =  = 223.\r\n" );
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document.write( "4.  Now we will find b, c and d in one line each:\r\n" );
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document.write( "    b =  =  = 279;\r\n" );
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document.write( "    c =  =  = 203;   and\r\n" );
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document.write( "    d =  =  = 261.\r\n" );
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document.write( "So, there are two answers (two solutions):\r\n" );
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document.write( "    a) a =  223;  b = 279;  c =  203;  d =  261;  a*b*c*d = 3296443311 and a+b+c+d =  966;\r\n" );
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document.write( "    b) a = -223;  b =-279;  c = -203;  d = -261;  a*b*c*d = 3296443311 and a+b+c+d = -966.\r\n" );
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