Okay, let's see. The question is asking: For how many two-digit primes is the sum of the digits equal to 8? Hmm, so I need to find all two-digit prime numbers where when you add the tens digit and the units digit together, the result is 8. Then count how many such primes there are. Alright, let's break this down step by step.

First, let's recall that a two-digit number can be represented as 10a + b, where a is the tens digit (ranging from 1 to 9) and b is the units digit (ranging from 0 to 9). The sum of the digits is a + b, which needs to equal 8. So, we need all primes where a + b = 8.

But wait, not all primes will have digits that sum to 8. So, the first step is to list all two-digit numbers where a + b = 8, and then check which of those numbers are prime. Alternatively, we can think of generating all two-digit primes first and then check if their digits sum to 8. Either approach would work, but maybe the first approach is more efficient since there are fewer possibilities for a + b = 8.

Let me try the first approach. Let's find all two-digit numbers where the sum of the digits is 8. Since a and b are digits, a can be from 1 to 9, and b is from 0 to 9, with a + b = 8. So, possible combinations are:

a = 1, b = 7 → 17

a = 2, b = 6 → 26

a = 3, b = 5 → 35

a = 4, b = 4 → 44

a = 5, b = 3 → 53

a = 6, b = 2 → 62

a = 7, b = 1 → 71

a = 8, b = 0 → 80

Wait, but 80 is a two-digit number, but let's check if these are all. Let's list them:

17, 26, 35, 44, 53, 62, 71, 80.

So, these are all the two-digit numbers where the sum of the digits is 8. Now, from these, we need to find which ones are prime numbers.

Let's check each number one by one.

1. 17: Is 17 a prime? Yes, 17 is a prime number. Its divisors are 1 and 17. So, 17 is prime.

2. 26: Is 26 a prime? No. 26 is divisible by 2 and 13. So, 26 is not prime.

3. 35: Is 35 a prime? No. 35 is divisible by 5 and 7. So, not prime.

4. 44: Is 44 a prime? No. 44 is divisible by 2, 4, 11, 22. Definitely not prime.

5. 53: Is 53 a prime? Yes. Let's check. 53 is a well-known prime number. Divisors are 1 and 53. So, prime.

6. 62: Is 62 a prime? No. 62 is divisible by 2 and 31. Not prime.

7. 71: Is 71 a prime? Yes. 71 is a prime number. Divisors are 1 and 71. Prime.

8. 80: Is 80 a prime? No. 80 is divisible by 2, 5, 8, 10, etc. Definitely not prime.

So, from the list, the primes are 17, 53, and 71. Let's count them: 17 is one, 53 is two, 71 is three. So, that's three primes. Wait, but let me double-check to make sure I didn't miss any or include any extra.

Wait, 17: correct, 53: correct, 71: correct. 80 is not prime, 26, 35, 44, 62 are all composite. So, total of 3 primes.

But hold on, let me verify each step again. Maybe I missed a two-digit number where the digits sum to 8. Let's see.

The possible two-digit numbers with digits adding up to 8 are:

Starting from 10: 1 + 7 = 8 → 17

20: 2 + 6 = 8 → 26

30: 3 + 5 = 8 → 35

40: 4 + 4 = 8 → 44

50: 5 + 3 = 8 → 53

60: 6 + 2 = 8 → 62

70: 7 + 1 = 8 → 71

80: 8 + 0 = 8 → 80

90: 9 + (-1) = 8 → Wait, that's not possible. So, 90 would require a units digit of -1, which isn't a digit. So, 90 is not in the list. So, the numbers are indeed 17, 26, 35, 44, 53, 62, 71, 80. That's 8 numbers. Then we checked each for primality and found 3 primes. So, the answer should be 3.

But wait, let me check if there's any other two-digit prime where the sum of digits is 8. Maybe I missed a number. For example, 101 is a prime, but it's three digits. So, not relevant here. Let's confirm all two-digit primes and check their digit sums.

Alternatively, list all two-digit primes and then check their digit sums. Maybe that's another way to approach.

The two-digit primes start from 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

Now, let's compute the sum of digits for each:

11: 1+1=2

13: 1+3=4

17: 1+7=8 → this is one.

19: 1+9=10

23: 2+3=5

29: 2+9=11

31: 3+1=4

37: 3+7=10

41: 4+1=5

43: 4+3=7

47: 4+7=11

53: 5+3=8 → two.

59: 5+9=14

61: 6+1=7

67: 6+7=13

71: 7+1=8 → three.

73: 7+3=10

79: 7+9=16

83: 8+3=11

89: 8+9=17

97: 9+7=16

So, the primes with digit sum 8 are 17, 53, and 71. That's three primes. So, this confirms the previous result. So, the answer is 3.

But wait, let me make sure I didn't miss any primes. Let's check again.

Looking at the list of two-digit primes:

11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

Sum of digits:

11: 2

13: 4

17:8

19:10

23:5

29:11

31:4

37:10

41:5

43:7

47:11

53:8

59:14

61:7

67:13

71:8

73:10

79:16

83:11

89:17

97:16

So, the primes with digit sum 8 are 17, 53, 71. That's three. So, the answer is 3.

Therefore, the number of two-digit primes where the sum of the digits is equal to 8 is 3.
</think>