<Question>
For how many two-digit primes is the sum of the digits equal to 8?
</Question>

<think>
<Plan_1>
Okay, so I need to figure out how many two-digit prime numbers have digits that add up to 8. Let me start by understanding the problem step by step.

<prompt_1.1> Understand the problem and outline the steps needed to solve it. </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
First, the question is about two-digit primes. That means numbers from 10 to 99, right? And each of these primes must have their individual digits sum up to exactly 8. For example, if the number is 17, the digits 1 and 7 add up to 8. But 17 is a prime number, so that would count. Wait, actually, 17 is a prime, and 1+7=8, so that's one example. But I need to find all such primes.

So, the plan is: first, list all two-digit numbers where the sum of the digits is 8, and then check which of those numbers are prime.
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
That seems manageable.

Let me start by listing all two-digit numbers where the digits add up to 8. Let's denote the two-digit number as AB, where A is the tens digit and B is the units digit. Then, A + B = 8. Since it's a two-digit number, A can range from 1 to 9, and B from 0 to 9. But since A + B = 8, A can be from 1 to 8, and B would be 8 - A.

So, let's list them:

<prompt_2.1> List all two-digit numbers where the sum of the digits is 8 based on the above plan. </prompt_2.1>
</Plan_2>

<Execution_2>
<execution_2.1>
- 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

So these are the numbers: 17, 26, 35, 44, 53, 62, 71, 80.
</execution_2.1>
</Execution_2>

<Plan_3>
Based on execution_2.1:
Now, from these numbers, I need to pick out the prime numbers. Let's check each one:

<prompt_3.1> Check which numbers from the list are prime based on the above plan. </prompt_3.1>
-----

Wait, let me double-check that I didn't miss any numbers. Let's see: the sum of digits is 8, so starting from 17, which is prime.

<prompt_3.2> Double-check the list of numbers and their primality based on the above plan. </prompt_3.2>
</Plan_3>

<Execution_3>
<execution_3.1>
1. 17: Prime? Yes, 17 is a prime number. Divisible only by 1 and itself.
2. 26: Even number, divisible by 2 and 13. Not prime.
3. 35: Ends with 5, divisible by 5 and 7. Not prime.
4. 44: Even, divisible by 2. Not prime.
5. 53: Prime? Let's see. 53 is a prime number. It's not divisible by 2, 3 (5+3=8, not divisible by 3), 5 (doesn't end with 0 or 5), 7 (7*7=49, 7*8=56, so no), 11 (11*4=44, 11*5=55, so no). So yes, prime.
6. 62: Even number, divisible by 2. Not prime.
7. 71: Prime? Check divisibility. 71 divided by 2? No. 3? 7+1=8, not divisible by 3. 5? Ends with 1, so no. 7? 7*10=70, 7*10+1=71, so 7*10=70, 71-70=1, so remainder 1. Not divisible by 7. 11? 11*6=66, 11*7=77, so no. So 71 is prime.
8. 80: Even number, divisible by 2 and 5. Not prime.

So from the list, 17, 53, and 71 are primes. That makes three numbers.
</execution_3.1>

<execution_3.2>
Then 26, not prime. 35, not. 44, not. 53, prime. 62, not. 71, prime. 80, not. So that's three primes.

But wait, let me make sure I didn't miss any other two-digit numbers where digits add to 8. For example, is there a number where A=9? Then B would be 8 - 9 = -1, which isn't possible, so A can't be 9. So the list is correct as above.
</execution_3.2>
</Execution_3>

<Plan_4>
Based on execution_3.1:
Wait a second, let me verify each of these.

<prompt_4.1> Verify the primality of the numbers identified as primes based on the above plan. </prompt_4.1>
-----

Based on execution_3.2:
Another way to check: maybe list all primes between 10 and 99 and then check which ones have digits summing to 8.

<prompt_4.2> Cross-verify by listing all primes between 10 and 99 and checking their digits based on the above plan. </prompt_4.2>
</Plan_4>

<Execution_4>
<execution_4.1>
17: Prime, correct.

26: 26/2=13, so nope.

35: 35/5=7, so nope.

44: Even, so nope.

53: As above, seems prime.

53: 5+3=8, yes. Prime.

71: Prime, yes.

71: 7+1=8, yes. Prime.

80: Even, ends with 0, divisible by 10, so nope.

So 17, 53, 71. Three primes.
</execution_4.1>

<execution_4.2>
But that might take longer, but let's try.

The primes between 10 and 99 are:

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

Now, check which of these have digits summing to 8.

11: 1+1=2 → no
13: 1+3=4 → no
17: 1+7=8 → yes
19: 1+9=10 → no
23: 2+3=5 → no
29: 2+9=11 → no
31: 3+1=4 → no
37: 3+7=10 → no
41: 4+1=5 → no
43: 4+3=7 → no
47: 4+7=11 → no
53: 5+3=8 → yes
59: 5+9=14 → no
61: 6+1=7 → no
67: 6+7=13 → no
71: 7+1=8 → yes
73: 7+3=10 → no
79: 7+9=16 → no
83: 8+3=11 → no
89: 8+9=17 → no
97: 9+7=16 → no

So again, we get 17, 53, 71. Three primes. That confirms the previous result.
</execution_4.2>
</Execution_4>

<Plan_5>
Based on execution_4.1:
But wait, let me check if I missed any primes.

<prompt_5.1> Recheck the list of numbers and their primality based on the above plan. </prompt_5.1>
-----

Based on execution_4.2:
Therefore, the answer is 3.

But just to be thorough, let me check each prime again:

<prompt_5.2> Perform a final verification of the solution based on the above plan. </prompt_5.2>
</Plan_5>

<Execution_5>
<execution_5.1>
Wait, 80 is the last one, but maybe I missed a number? Let's see:

Wait, when 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. That's all the combinations where digits add to 8. So 8 numbers total. Then of those, three are primes. So the answer is 3.
</execution_5.1>

<execution_5.2>
17: digits 1 and 7, sum 8. Prime. Yes.

53: digits 5 and 3, sum 8. Prime. Yes.

71: digits 7 and 1, sum 8. Prime. Yes.

No others in the list. So three primes.
</execution_5.2>
</Execution_5>

<Plan_6>
Based on execution_5.2:
**Final Answer**
The number of two-digit primes with digits summing to 8 is \boxed{3}.
</Plan_6>

<Execution_6>
No prompt to execute.
</Execution_6>

<Final_answer>
Thus, there are $\boxed{3}$ two-digit primes for which the sum of the digits is equal to $8$.
</Final_answer>
</think>