You'll need a list of 2-digit prime numbers, shown below: 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97
The first one on the list that qualifies is 17, because 1+7=8. So check out the rest on the list. I find a total of three, including 17.
First, we list all the two-digit numbers with a digits sum of 8:
17, 26, 35, 44, 53, 62, 71, 80
Clearly, 26, 44, 62, and 80 are not prime. 35 is not prime, but 17, 53, and 71 are (we can verify this by dividing each number by all the primes less than its square root.
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