Find all integer solutions to: a + b + c = 24, a2 + b2 + c2 = 210, abc = 440.
Solution
ab + bc + ca = ( (a + b + c)2 - (a2 + b2 + c2) )/2 = 183, so a, b, c are roots of the cubic x3 - 24x2 + 183x - 440 = 0. But it easily factorises as (x - 5)(x - 8)(x - 11) = 0, so the only solutions are permutations of (5, 8, 11).
© John Scholes
jscholes@kalva.demon.co.uk
1 July 2002