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The smallest Prime number, of course, is 2.

The smallest prime number whose digits add up to 2 is, of course, 2.

The next-smallest is 11.

It is left as an exercise for the reader to determine whether the other numbers whose digits add up to 2 (e.g. 101, 1001, 10001) are prime.

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Q: What is the prime number that its digits adds up to the smallest prime number?
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