<Question>
How many positive whole-number divisors does 196 have?
</Question>

<think>
<Plan_1>
Okay, so I need to figure out how many positive whole-number divisors 196 has. Hmm, let's start by recalling that the number of divisors a number has is related to its prime factorization. I remember something about breaking the number down into primes and then using exponents to calculate the total number of divisors. Let me try to remember the exact method.

First, I should find the prime factors of 196. Let's start dividing by the smallest primes.

<prompt_1.1> Execute the above plan to find the prime factors of 196. </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
196 is an even number, so it's divisible by 2. 

196 ÷ 2 = 98. Okay, so 2 is a prime factor. Now, 98 is also even, so divide by 2 again: 98 ÷ 2 = 49. So far, the prime factors are 2, 2, and then 49. Now, 49 is left. 49 is 7 squared, right? So 7 × 7 = 49. Therefore, the prime factors of 196 are 2² × 7².
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Now, I think the formula for finding the number of divisors is to take the exponents of the prime factors, add 1 to each of them, and then multiply those together. Let me verify that. So if a number N can be expressed as N = p₁^a × p₂^b × p₃^c × ... then the number of divisors is (a+1)(b+1)(c+1)... etc. Yeah, that sounds right. Because for each prime, the exponent can range from 0 up to the maximum exponent in the prime factorization.

<prompt_2.1> Execute the above plan to calculate the number of divisors of 196. </prompt_2.1>
-----

But let me also try to list them out to make sure I didn't make a mistake. Starting from 1, which is a divisor of every number. Then 2, 7, 14 (2×7), 49 (7×7), 98 (2×49), and 196 itself. Let me check each one:

<prompt_2.2> Execute the above plan to list and verify all divisors of 196. </prompt_2.2>
</Plan_2>

<Execution_2>
<execution_2.1>
For example, for 2², the exponents can be 0, 1, or 2. That's 3 choices. For 7², the exponents can be 0, 1, or 2 as well. That's another 3 choices. So multiplying 3 × 3 gives 9. Therefore, 196 should have 9 positive divisors.
</execution_2.1>

<execution_2.2>
1 divides 196.
2 divides 196 (since it's even).
7 divides 196 (since 7×28=196).
14 divides 196 (2×7=14, and 196 ÷14=14).
Wait, 14×14=196, so 14 is a divisor.
Then 49 divides 196 (7×28=196, so 49 is a divisor).
98 divides 196 (2×49=98, and 196 ÷98=2).
196 divides itself.

So the divisors are: 1, 2, 7, 14, 49, 98, 196. That's 7 divisors.
</execution_2.2>
</Execution_2>

<Plan_3>
Based on execution_2.1:
Wait, let me check that again. Let's list them out to confirm.

<prompt_3.1> Execute the above plan to list and count the divisors of 196. </prompt_3.1>

But wait, maybe I missed some? Let's see. Let's list all possible combinations.

<prompt_3.2> Execute the above plan to verify the list of divisors of 196. </prompt_3.2>

Hmm, but I only get 7 when I list them. That's conflicting with the formula result of 9.
Where did I go wrong?

Wait, maybe I missed some divisors when listing them. Let me try again.

<prompt_3.3> Execute the above plan to resolve the discrepancy between the formula and manual listing of divisors. </prompt_3.3>

But why did the formula give me 9? Let me check the formula again.

<prompt_3.4> Execute the above plan to verify the formula for calculating the number of divisors. </prompt_3.4>

Let me list all combinations again:

<prompt_3.5> Execute the above plan to list all combinations of divisors of 196. </prompt_3.5>

But why did I think there were only 7 when I listed them before? Let me check again.

<prompt_3.6> Execute the above plan to resolve the discrepancy in the number of divisors of 196. </prompt_3.6>

But just to be thorough, let's list all the divisors:

<prompt_3.7> Execute the above plan to list all divisors of 196 and confirm the count. </prompt_3.7>

But why did I think there were only 7 when I listed them before? Let me check again.

<prompt_3.8> Execute the above plan to resolve the discrepancy in the number of divisors of 196. </prompt_3.8>
</Plan_3>

<Execution_3>
<execution_3.1>
Starting with 1, 2, 7. Then 14 (2×7). Then 4 (but 4 doesn't divide 196 because 196 ÷4 is 49, which is 49, but 4 is 2². Wait, 2² is a factor, so 4 should divide 196. Wait, 196 ÷4 = 49, which is an integer. So 4 is a divisor. Then 14 (2×7), 28 (4×7), 49 (7²), 98 (14×7), and 196. Let me list them properly:

1, 2, 4, 7, 14, 28, 49, 98, 196. That's 9 divisors.
</execution_3.1>

<execution_3.2>
The exponents for 2 are 0, 1, 2. The exponents for 7 are 0, 1, 2. So combining them:

2^0 × 7^0 = 1

2^0 × 7^1 = 7

2^0 × 7^2 = 49

2^1 × 7^0 = 2

2^1 × 7^1 = 14

2^1 × 7^2 = 98

2^2 × 7^0 = 4

2^2 × 7^1 = 28

2^2 × 7^2 = 196

Wait, hold on, I must have missed 4 and 28. So the divisors are: 1, 2, 4, 7, 14, 28, 49, 98, 196. That's 9 divisors. So earlier, when I tried listing them, I missed 4 and 28.
</execution_3.2>

<execution_3.3>
Starting with 1, 2, 7, 14, 49, 98, 196. Hmm, that's 7. But according to the formula, it should be 9. What's missing here?
</execution_3.3>

<execution_3.4>
If 196 is 2² × 7², then the exponents for 2 can be 0, 1, or 2. For 7, same thing: 0, 1, or 2. So the possible combinations are:

2^0 × 7^0 = 1
2^0 × 7^1 = 7
2^0 × 7^2 = 49
2^1 × 7^0 = 2
2^1 × 7^1 = 14
2^1 × 7^2 = 98
2^2 × 7^0 = 4
2^2 × 7^1 = 28
2^2 × 7^2 = 196

Wait, 28? Wait, 2² × 7^1 is 4 × 7 = 28. But 28 divides 196? Let me check: 196 ÷ 28 = 7. Yes, because 28 × 7 = 196. So I must have missed 4 and 28 when I listed them before.

So the complete list of divisors is: 1, 2, 4, 7, 14, 28, 49, 98, 196. That's 9 divisors. So I must have missed some when I did the initial list.
</execution_3.4>

<execution_3.5>
The exponents for 2 are 0, 1, 2. The exponents for 7 are 0, 1, 2. So all combinations:

2^0 × 7^0 = 1
2^0 × 7^1 = 7
2^0 × 7^2 = 49
2^1 × 7^0 = 2
2^1 × 7^1 = 14
2^1 × 7^2 = 98
2^2 × 7^0 = 4
2^2 × 7^1 = 28
2^2 × 7^2 = 196

Ah, there we go! I must have missed some when I was listing earlier. So the divisors are: 1, 2, 4, 7, 14, 28, 49, 98, 196. That's 9 divisors.
</execution_3.5>

<execution_3.6>
Wait, 196 is 14 squared. 14 is 2×7, so 14 squared is 2²×7². So the divisors should be all combinations of 2^a * 7^b where a is 0,1,2 and b is 0,1,2. Let's list them systematically:

For a=0:
7^0 = 1
7^1 = 7
7^2 = 49

For a=1:
2^1 * 7^0 = 2
2^1 * 7^1 = 14
2^1 * 7^2 = 98

For a=2:
2^2 * 7^0 = 4
2^2 * 7^1 = 28
2^2 * 7^2 = 196

Wait, wait! I missed 4 and 28 and 196. So the full list is:

1, 7, 49,
2, 14, 98,
4, 28, 196.

That's 9 divisors. So earlier, when I listed them, I must have missed 4 and 28. Let me check:

Starting from 1, 2, 7, 14, 49, 98, 196. But where are 4 and 28? Oh, right, because 4 divides 196? Let's check 196 ÷4=49, which is an integer. So yes, 4 is a divisor. Similarly, 28 divides 196? 196 ÷28=7, which is an integer. So 28 is also a divisor. So the complete list is:

1, 2, 4, 7, 14, 28, 49, 98, 196. That's 9 divisors. So I must have missed 4 and 28 when I was listing them the first time.
</execution_3.6>

<execution_3.7>
1) 1
2) 2
3) 7
4) 14 (2×7)
5) 49 (7²)
6) 98 (2×49)
7) 196 (2²×7²)

Wait, that's still 7. Hmm. What am I missing here? 

Wait, maybe there's another divisor. Let's see. Let's check 196 divided by 1, 2, 7, 14, 49, 98, 196. Are there any numbers in between that divide 196? Let's check 4. 196 ÷4=49. So 4 is a divisor. Then 49 is already listed. Then 7 is there, 2, 14, 98, 196. Wait, 4 is a divisor. Let me check 49. 49 is 7². So 49 is a divisor. Then 2×49=98. Then 7×14=98. Wait, but 14 is 2×7, which is already there. Wait, maybe I missed 4 and 49? Wait, 4 is 2², which is a factor. So 4 is a divisor. Then 49 is 7². So those are two more divisors. So adding 4 and 49. So the list should be:

1, 2, 4, 7, 14, 49, 98, 196. Wait, that's 8. But according to the formula, it should be 9. What's the ninth one?
</execution_3.7>

<execution_3.8>
Wait, maybe I made a mistake in the prime factorization? Let me double-check that. 196 divided by 2 is 98, divided by 2 again is 49, which is 7 squared. So 2² × 7². That seems correct. Then exponents for 2 are 0,1,2; for 7 are 0,1,2. So combinations:

2^0 * 7^0 = 1

2^0 * 7^1 = 7

2^0 * 7^2 = 49

2^1 * 7^0 = 2

2^1 * 7^1 = 14

2^1 * 7^2 = 98

2^2 * 7^0 = 4

2^2 * 7^1 = 28

2^2 * 7^2 = 196

Ah! There we go. I missed 4 and 28 in my previous listing. So the divisors are: 1, 2, 4, 7, 14, 28, 49, 98, 196. That's 9 divisors. So my initial listing was incomplete. Therefore, the formula was correct, and there are indeed 9 divisors.
</execution_3.8>
</Execution_3>

<Plan_4>
Based on execution_3.1, execution_3.2, execution_3.3, execution_3.4, execution_3.5, execution_3.6, execution_3.7, execution_3.8:
The answer is 9. So I think that's the correct answer. Let me just recap to make sure I didn't make a mistake.

<prompt_4.1> Recap and confirm the final answer for the number of divisors of 196. </prompt_4.1>
</Plan_4>

<Execution_4>
<execution_4.1>
Prime factorization: 2² ×7². Number of divisors: (2+1)(2+1)=9. Listing them out gives 1, 2, 4, 7, 14, 28, 49, 98, 196. That's 9 numbers. All check out when dividing 196. So yes, the answer is 9.
</execution_4.1>
</Execution_4>

<Final_answer>
Therefore, the number of positive whole-number divisors of 196 is $\boxed{9}$.
</Final_answer>
</think>