SOLUTION: 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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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=?
Found 2 solutions by MathLover1, ikleyn:
Answer by MathLover1(20850)   (Show Source): You can put this solution on YOUR website!

,
,
,
,
,

solve this:
=? ,

since given and , we have

now, we will find out what are ,,,and equal to:

->.....1)
->....1)
from 1) and 2):
=>=>=>=>....3)

since we have ....4)

from 3) and 4) we have
............solve for





find :
....3)




go to , plug in and find




go to ,plug in and find





so, we have:





and, finally we can find

check the product:
....confirmed

Answer by ikleyn(52788)   (Show Source): You can put this solution on YOUR website!
.
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=?
~~~~~~~~~~~~~~~~~~~~~~~~

I will show you more simple and more straightforward solution. 

0.  First of all,  EITHER  all four numbers a, b, c and d are POSITIVE, OR all four of them are NEGATIVE.

    Indeed, had two numbers a and b, for example, have different signs, their product ab would be negative.

    Since all given values of products are positive, the statement made is true.

    It means that there are two solutions:

         one solution with all four numbers positive, and
         the second solution with all four numbers negative (opposite).

    For simplicity, let us start finding positive solution, and 
        then will take the opposite numbers as the second solution, for completeness.



1.  The most easy question is about the product a*b*c*d. 

    It is equal  a*b*c*d = (a*b)*(c*d) = 62217*52983 = 3296443311.   (1)



2.  Now I want to find the value of "a".

    For it, I will collect all given products that DO NOT CONTAIN the multiplier (the factor) "a".

    b*c = 56637,    (2)

    c*d = 52983,    (3)

    b*d = 72819.    (4)


Now multiply  all three equations (2), (3) and (4) (both sides).  You will get
    
     = 56637*52983*72819,   or

     = 218515122014049,    which implies (after taking square root of both sides)

    b*c*d = 14782257     (since we are looking now for positive numbers only).


    Then a =  =  = 223.


4.  Now we will find b, c and d in one line each:

    b =  =  = 279;

    c =  =  = 203;   and

    d =  =  = 261.


So, there are two answers (two solutions):

    a) a =  223;  b = 279;  c =  203;  d =  261;  a*b*c*d = 3296443311 and a+b+c+d =  966;

    b) a = -223;  b =-279;  c = -203;  d = -261;  a*b*c*d = 3296443311 and a+b+c+d = -966.

Solved.


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