Competition problem — Number theory

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Number theory Difficulty 4.5 AIME Find the answer Romania

How many three-digit primes can be turned into perfect cubes with a permutation of their digits?

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

The three-digit perfect cubes are $125$ , $216$ , $343$ , $512$ and $729$ . The primes must end with an odd digit, different from $5$ , and must be not divisible by $3$ . Therefore we must consider only $251$ , $521$ , $433$ . All of them are primes.

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