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Find the positive integer solutions of the equation (1+1/a)(1+1/b)(1+1/c) = 2

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Dodgy Proofs

What is wrong with these dodgy proofs?

Code to Zero

Age 16 to 18 Challenge Level:

Find all the 3 digit numbers such that by adding the first digit, the square of the second digit and the cube of the third digit you get the original number, for example $1 + 3^2 + 5^3 = 135$. Can you find ways of reducing the search so that you do not need to test 900 cases?

[This problem appears in Ken Follett's spy thriller of the same name].