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The two-digit number that is the square of its digit sum

numbersdigitssquaressearch

Find every two-digit number that is equal to the square of the sum of its own digits.

For example, 25 has digit sum 7, and 7 squared is 49, not 25, so 25 does not work. Explain why the search is short, and name the analogous question for cubes.

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The number is a perfect square between 10 and 99, so its digit sum is one of 4, 5, 6, 7, 8, 9. Check those six squares.

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