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A square is a cube (Posted on 2012-11-29) Difficulty: 1 of 5
What 2-digit number(s), when squared, produces a number equal to the cube of its sum of digits?

No Solution Yet Submitted by Ady TZIDON    
Rating: 3.0000 (1 votes)

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Answer | Comment 1 of 2
27 is the only such number.

My method:
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
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