Competition problem — Number theory

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Number theory Difficulty 5.2 AIME, harder Prove it 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.

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 $30$ .

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