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}}}=? , 


since given {{{a*b=62217}}} and  {{{c*d= 52983}}}, we have

{{{A*B*C*D=(A*B)*(C*D) =62217*52983=3296443311}}}

now, we will find out what are {{{A}}},{{{B}}},{{{C}}},and {{{D}}} equal to:


{{{a*b=62217}}}->{{{b=62217/a}}}.....1)
{{{b*d=72819}}}->{{{b=72819/d}}}....1)

from 1) and 2):
=>{{{62217/a=72819/d}}}=>{{{d/a=72819/62217}}}=>{{{d/a=261/223}}}=>{{{d=261a/223}}}....3)


since {{{d*a=58203}}} we have {{{d=58203/a}}}....4)


from 3) and 4) we have

{{{261a/223 =58203/a}}}............solve for {{{a}}}

{{{a*a/223 =58203/261}}}

{{{a^2=(58203/261)223 }}}

{{{a^2=(223)223 }}}

{{{a^2=(223)^2 }}}

{{{highlight(a=223 )}}}

find {{{d}}}:

{{{d=261a/223}}}....3)

{{{d=(261*cross(223))/cross(223)}}}

{{{highlight(d=261)}}}



go to  {{{a*b=62217}}}, plug in {{{a}}} and find {{{b}}}

 {{{223*b=62217}}}
 {{{b=62217/223}}}

{{{highlight(b=279)}}}


go to {{{b*c=56637}}},plug in {{{b}}} and find {{{c}}}

{{{279*c=56637}}}

{{{c=56637/279}}}

{{{highlight(c=203)}}}



so, we have:
 {{{A=highlight(a=223 )}}}
 
{{{B=highlight(b=279)}}}

{{{C=highlight(c=203)}}}

{{{D=highlight(d=261)}}}

and, finally we can find  {{{a+b+c+d}}}

{{{a+b+c+d=223+279+203+261=966}}}

check the product:

{{{A*B*C*D=223*279*203*261=3296443311}}}....confirmed