unable to find a rational soln

a+b=8 & a^3 +b^3=152...a=3, b=5
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the way to arrive at the solution is equally important.. Method I --------------------- Here a+b = 8 a3 +b3 can be written as (a+b)3 -3ab(a+b) 152 = 8^3 - 3ab(8) solving, ab=15 a-b = +/- SqRoot((a+b)^2 -4ab)) => a-b = +/-2 solving a = 5 or 3; b= 3 or 5 hence the numbers are 3 and 5 ----------------------------------------------------- Method 2 a, b are natural numbers whose sum is less than 8. a3 + b3 =152. the largest cube less than 152 is 125 ; (5)^3 Further sum of two even or two odd nos. is even hence a and b both should be even or odd. the a and b can be two of 5,3,1, but a or b !=1 since 151 is not a perfect cube. or a or b = 2,4 but 4 ^3 = 64 way less than 152 hence not possible the only ans is 5 and 3
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