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Problem 1215

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Number theory Difficulty 5.2 Prove it Ukrainian National Mathematical Olympiad, 3rd Round · Ukraine

Find all natural numbers $N$ , that have two digits and are equal to the sum of their digits added to the cube of this sum.

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

Let $n$ be the sum of the digits, then: $N = n + n^3$ . The cube has to be less than $99$ , that gives us four options: $1$ , $2$ , $3$ , $4$ . An easy check shows that the only possible answer is $30$ .

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.