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To find the smallest possible sum of roots for any product, the answer is always whatever numbers are closest together. This fact can be derived by the fact that the greatest area we can enclose with a given length is a perfect square. So we can take the square root of 7272 and use that, since they will be exactly the same number.
If you are looking for whole numbers, you would have to go up or down until you find a factor of 7272 closest to the square root. In this case that would be 72 and 101.