<Question>
Suppose $a$ and $b$ are different prime numbers greater than 2. How many whole-number divisors are there for the integer $a(2a+b)-2a^{2}+ab$?
</Question>

<think>
<Plan_1>
Okay, let me try to figure out this problem. So, the question is: If a and b are different prime numbers greater than 2, how many whole-number divisors does the integer a(2a + b) - 2a² + ab have? Hmm, first, I need to simplify the expression given. Let me write it down step by step.

<prompt_1.1> Simplify the expression a(2a + b) - 2a² + ab as per the above plan. </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
The expression is a(2a + b) - 2a² + ab. Let me expand the first term. a times (2a + b) would be 2a² + ab. So, substituting that back into the expression, we have 2a² + ab - 2a² + ab. Wait, let me check that again. The original expression is a(2a + b) which is 2a² + ab, then subtract 2a² and add ab. So combining those terms: 2a² + ab - 2a² + ab. The 2a² and -2a² cancel each other out, right? So that leaves ab + ab, which is 2ab. So the entire expression simplifies to 2ab.
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Wait, that seems straightforward. Let me verify once more.
Original expression: a(2a + b) - 2a² + ab. Expand the first term: 2a² + ab - 2a² + ab. Yes, the 2a² terms cancel, leaving 2ab. So the value of the expression is 2ab.

Now, the question is asking for the number of whole-number divisors of this integer, which is 2ab. Since a and b are different prime numbers greater than 2, they are both odd primes. For example, 3, 5, 7, etc. So, 2 times a times b would be 2ab, where a and b are distinct odd primes.

To find the number of divisors, I need to factorize the number into its prime factors and use the formula for the number of divisors. The formula is that if a number N has the prime factorization N = p₁^e₁ * p₂^e₂ * ...

<prompt_2.1> Determine the number of divisors of 2ab using the formula for the number of divisors as per the above plan. </prompt_2.1>
-----

Wait, but let me make sure there's no catch here. The problem states that a and b are different prime numbers greater than 2.
So, they are distinct odd primes. Therefore, 2ab has prime factors 2, a, and b, each with exponent 1, except for 2 which has exponent 1 as well.

<prompt_2.2> Re-evaluate the factorization and number of divisors of 2ab considering the properties of a and b as per the above plan. </prompt_2.2>
-----

But let me check with an example to make sure. Let's take the smallest primes greater than 2, which are 3 and 5.

<prompt_2.3> Verify the solution with specific examples using primes 3 and 5, then 3 and 7, and so on as per the above plan. </prompt_2.3>
</Plan_2>

<Execution_2>
<execution_2.1>
* , then the number of divisors is (e₁ + 1)(e₂ + 1)... (eₙ + 1). 

So, let's factorize 2ab. Since a and b are primes, 2ab is already the product of primes: 2, a, and b. Each of these primes is distinct because a and b are different primes. Therefore, the exponents for each prime factor are all 1. 

Therefore, the number of divisors would be (1 + 1)(1 + 1)(1 + 1) = 2*2*2 = 8. So, 8 divisors.
</execution_2.1>

<execution_2.2>
So the exponents are all 1. Therefore, the number of divisors is (1+1)(1+1)(1+1) = 2*2*2 = 8. So, 8 divisors. Therefore, the answer should be 8.
</execution_2.2>

<execution_2.3>
Let's compute the expression for a=3 and b=5. Then 2ab = 2*3*5 = 30. The divisors of 30 are 1,2,3,5,6,10,15,30. That's 8 divisors. Now, let's check if the expression evaluates to 30. Original expression: a(2a + b) - 2a² + ab. Plugging in a=3, b=5: 3*(6 +5) - 2*9 +3*5 = 3*11 -18 +15 =33 -18 +15= 33 -18 is 15, 15+15=30. Correct. So 30 has 8 divisors.

Another example: a=3, b=7. Then 2ab=2*3*7=42. Divisors of 42: 1,2,3,6,7,14,21,42. Again 8 divisors. Let's check the expression: 3*(6 +7) -2*9 +3*7=3*13 -18 +21=39 -18=21 +21=42. Correct. So 8 divisors.

Another example: a=5, b=7. 2ab=70. Divisors of 70:1,2,5,7,10,14,35,70. 8 divisors. The expression:5*(10 +7) -2*25 +5*7=5*17=85 -50=35 +35=70. Correct. So again 8 divisors.
</execution_2.3>
</Execution_2>

<Plan_3>
Based on execution_2.1:
Wait, hold on. Let me double-check if I missed anything. The expression simplifies to 2ab, and since a and b are distinct primes greater than 2, they are co-prime to each other and to 2. So 2ab has prime factors 2, a, and b, each appearing once.
Therefore, the number of divisors is indeed (1+1)(1+1)(1+1) = 8. So the answer should be 8.

But let me think again. Are there cases where a and b could be the same prime? Wait, the problem says a and b are different primes. So, a ≠ b.

<prompt_3.1> Confirm that the number of divisors is 8 considering that a and b are distinct primes as per the above plan. </prompt_3.1>
-----

But let me think if there's any other possibility. For example, what if a or b is 2?

<prompt_3.2> Confirm that the conditions of the problem ensure a and b are odd primes as per the above plan. </prompt_3.2>

Another check: suppose a=7, b=11.

<prompt_3.3> Perform a final verification with additional examples to ensure the solution is consistent as per the above plan. </prompt_3.3>
</Plan_3>

<Execution_3>
<execution_3.1>
Therefore, 2ab is the product of three distinct primes: 2, a, and b. Hence, the number of divisors is (1+1)(1+1)(1+1) = 8. Therefore, the number of whole-number divisors is 8.
</execution_3.1>

<execution_3.2>
But the problem states that primes are greater than 2, so they can't be 2. So, a and b are odd primes, so 2 is a distinct prime from a and b. Therefore, 2ab has three distinct prime factors, each with exponent 1. Therefore, the number of divisors is (1+1)^3 = 8. 

Therefore, regardless of which primes a and b are (as long as they are distinct primes greater than 2), the expression 2ab will always have 8 divisors.
</execution_3.2>

<execution_3.3>
Then 2ab=154. Divisors:1,2,7,11,14,22,77,154. 8 divisors. Expression:7*(14 +11) -2*49 +7*11=7*25=175 -98=77 +77=154. Correct. So 8 divisors.

Wait, but in all these cases, a and b are odd primes, so 2ab is 2 times two odd numbers, hence even, but not divisible by 4, since a and b are odd. Therefore, 2ab has prime factors 2, a, b, each to the first power. Therefore, the number of divisors is 2*2*2=8.

Therefore, the answer is 8. So the number of whole-number divisors is 8.
</execution_3.3>
</Execution_3>

<Final_answer>
Therefore, there are $\boxed{8}$ whole-number divisors for the integer in question.
</Final_answer>
</think>