What 2-digit number(s), when squared, produces a number equal to the cube of its sum of digits?
27 is the only such number.
Call the digits a and b so the square is (10a+b)^2 and the cube is (a+b)^3
(10a+b)^2 = (a+b)^3
100a^2 + 20ab + b^2 = a^3 + 3a^2b + 3ab^2 + b^3
a^3 + (3b-100)a^2 + (3b^2 - 20b)a + (b^3 - b^2) = 0
Which is a cubic in a. While this is solvable in principle, in practice it was easier to make an array in Excel and see where the sum was zero.
Posted by Jer
on 2012-11-29 11:27:27