<Question>
If $A$ is the sum of the positive divisors of $500$, what is the sum of the distinct prime divisors of $A$?
</Question>

<think>
<Plan_1>
Okay, let's see. The problem is: If A is the sum of the positive divisors of 500, what is the sum of the distinct prime divisors of A? Hmm, right. So first, I need to find all the positive divisors of 500, add them up to get A, and then factorize A to find its distinct prime factors, and sum those primes. Alright, step by step.

First, let's recall how to find the sum of divisors of a number. I remember that if a number is expressed in its prime factorization form, say n = p₁^a₁ * p₂^a₂ * ... * p_k^a_k, then the sum of its divisors is (1 + p₁ + p₁² + ... + p₁^a₁) * (1 + p₂ + p₂² + ... + p₂^a₂) * ... * (1 + p_k + p_k² + ... + p_k^a_k). Yeah, that formula. So, I need to start by prime factorizing 500.

Let's do that.

<prompt_1.1> Prime factorize 500 and calculate the sum of its divisors using the formula. </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
500. Well, 500 is 5 * 100, and 100 is 2² * 5². So, putting that together: 500 = 5 * 2² * 5². Wait, that's 2² * 5³. Let me check: 2² is 4, 5³ is 125, and 4 * 125 is 500. Yep, that's correct. So the prime factors are 2 and 5, with exponents 2 and 3 respectively.

So using the sum of divisors formula, the sum A would be (1 + 2 + 2²) * (1 + 5 + 5² + 5³). Let me compute each part separately.

First, the sum for prime 2: 1 + 2 + 4 = 7. Then, the sum for prime 5: 1 + 5 + 25 + 125. Let's calculate that: 1 + 5 is 6, 6 +25 is 31, 31 +125 is 156. So, the total sum A is 7 * 156. Let me compute that. 7 * 150 is 1050, and 7 *6 is 42, so 1050 +42 = 1092. Therefore, A is 1092.
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Now, the next step is to find the distinct prime divisors of 1092 and sum them. So first, let's factorize 1092.

<prompt_2.1> Factorize 1092 to find its distinct prime divisors and calculate their sum. </prompt_2.1>
-----

Wait, but let me check if there's another way to approach this. Maybe I made a mistake in calculating the sum of divisors of 500. Let me verify that again.

<prompt_2.2> Re-verify the calculation of the sum of divisors of 500. </prompt_2.2>
</Plan_2>

<Execution_2>
<execution_2.1>
Let's start by dividing by 2: 1092 ÷ 2 = 546. 546 is even, so divide by 2 again: 546 ÷2 = 273. Now, 273 is odd. Let's check divisibility by 3: 2 +7 +3=12, which is divisible by 3. 273 ÷3 = 91. Then, 91: 9 +1=10, not divisible by 3. Let's see, 91 ÷7=13. Yes, because 7*13=91. And 13 is a prime. So putting it all together, the prime factors of 1092 are 2² * 3 * 7 *13. Therefore, the distinct prime divisors are 2, 3, 7, and 13. Adding them up: 2 +3 +7 +13 =25.
</execution_2.1>

<execution_2.2>
500's prime factorization is 2² * 5³. So sum of divisors is (1 + 2 + 4) * (1 + 5 + 25 + 125) = 7 * 156. 7*156: 7*150=1050, 7*6=42, 1050+42=1092. That's correct. So A is 1092.
</execution_2.2>
</Execution_2>

<Plan_3>
Based on execution_2.1:
Wait, hold on. Let me double-check my steps. Maybe I missed a prime factor when factorizing 1092? Let's go through the factorization again.

<prompt_3.1> Re-factorize 1092 to verify the prime factors and sum of distinct primes. </prompt_3.1>
-----

Based on execution_2.2:
Factorizing 1092: 1092 divided by 2 is 546, divided by 2 again is 273. 273 divided by 3 is 91.

<prompt_3.2> Confirm the sum of distinct prime divisors of 1092. </prompt_3.2>
</Plan_3>

<Execution_3>
<execution_3.1>
Starting with 1092: divide by 2 gives 546. Divide by 2 again gives 273. Then 273 divided by 3 is 91. Then 91 is 7*13. So primes are 2, 3, 7, 13. Yes, that's correct. So sum is 2+3+7+13=25.
</execution_3.1>

<execution_3.2>
Then 91 is 7*13. So primes are 2,3,7,13. Sum is 2+3+7+13=25.
</execution_3.2>
</Execution_3>

<Final_answer>
Therefore, the sum of the distinct prime divisors of $A$ is $\boxed{25}$.
</Final_answer>
</think>