Question 1208880
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If (a + 1)(b + 1)(a + b) = 1530
and a^3 + b^3 = 1241
then a + b = ?
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        It admits  (= it has)  fantastically brilliant solution.



<pre>
First,  (1 + a)(1 + b)(a + b) = (1 + a + b + ab)*(a+b) = (a+b) + (a+b)^2 + ab*(a+b).

Next,  a^3 + b^3 = (a+b)*(a^2 - ab + b^2) = (a+b)*((a+b)^2-3ab) = (a+b)^3 - 3ab*(a+b).



Thus you have these two equations

    (a+b) + (a+b)^2 + ab*(a+b)  = 1530    (1)

    (a+b)^3         - 3ab*(a+b) = 1241    (2)



Multiply equation (1) by 3; keep equation (2) as is.  Then add them.  You will get

    3(a+b) + 3(a+b)^2 + (a+b)^3 = 3*1530 + 1241,

or

    3(a+b) + 3(a+b)^2 + (a+b)^3 = 5831.



Add 1 to both sides

    1 + 3(a+b) + 3(a+b)^2 + (a+b)^3 = 5832.


Notice that the left side is ((a+b)+1)^3.


So, the last equation is

    ((a+b)+1)^3 = 5832.



Take cube root of both sides

    (a+b)+1 = {{{root(3,5832)}}} = 18.



From this, get

     a+b = 18-1 = 17.



<U>ANSWER</U>.  If (1 + a)(1 + b)(a + b) = 1530 and a^3 + b^3 = 1241,  then  a+b = 17.
</pre>

Solved.