Okay, let's see. I need to find the number of whole-number divisors of the integer given by the expression a(2a + b) - 2a² + ab, where a and b are different prime numbers greater than 2. Hmm, primes greater than 2, so they're odd primes. Let's start by simplifying the expression.

First, let me expand the terms step by step. The expression is a(2a + b) minus 2a² plus ab. Let me distribute the a into the parentheses:

a*2a = 2a²

a*b = ab

So the first term becomes 2a² + ab. Then subtract 2a² and add ab. Let me write that out:

2a² + ab - 2a² + ab

Wait, combining like terms here. The 2a² and -2a² cancel each other out. Then we have ab + ab, which is 2ab. So the entire expression simplifies to 2ab. That's much simpler!

So the integer we're looking at is 2ab, where a and b are different primes greater than 2. Since a and b are primes greater than 2, they must be odd. Let me confirm: primes greater than 2 are indeed odd primes, so a and b are both odd. Therefore, 2ab is twice the product of two odd primes. Let's check if this is correct.

Wait, let's verify the simplification again. Original expression:

a(2a + b) - 2a² + ab

Expanding a(2a + b) gives 2a² + ab. Then subtract 2a², so 2a² - 2a² = 0, and then add ab. So total is 0 + ab + ab = 2ab. Yes, that's correct. So the expression simplifies to 2ab.

Now, since a and b are different primes greater than 2, they are both odd, so ab is odd*odd = odd. Then 2ab is even. So 2ab is an even number.

To find the number of divisors, we need to factorize 2ab into its prime factors. Let's see. Since a and b are primes, and 2 is also a prime, the prime factorization of 2ab would be 2 * a * b. Since a and b are distinct primes, this factorization is unique.

Therefore, the prime factors are 2, a, and b. The number of divisors of a number is found by taking the exponents of its prime factors, adding 1 to each, and multiplying them together. In this case, the exponents are all 1 for each prime factor. So the number of divisors would be (1+1)(1+1)(1+1) = 2*2*2 = 8.

Wait, but hold on. Let me confirm. If the number is 2 * a * b, where a and b are distinct primes, then the number of divisors is indeed (1+1)(1+1)(1+1) = 8. For example, take 2*3*5=30. The divisors of 30 are 1, 2, 3, 5, 6, 10, 15, 30. That's 8 divisors. So yes, that seems right.

But wait, in the problem statement, a and b are different primes greater than 2. So they can't be 2. So in this case, 2 is a separate prime factor, a is another prime, and b is another prime. All distinct. So the prime factorization is indeed 2 * a * b, with exponents 1 each. Therefore, the number of divisors is 2*2*2=8.

Therefore, the integer 2ab has 8 divisors. So the answer should be 8.

But wait, let me double-check everything. Let me take specific primes to test. Let's choose a=3 and b=5. Then the expression is 2*3*5=30. The divisors of 30 are 1, 2, 3, 5, 6, 10, 15, 30. That's 8 divisors. Correct.

Another example: a=5, b=7. Then 2*5*7=70. Divisors of 70: 1, 2, 5, 7, 10, 14, 35, 70. Again 8. Seems consistent.

Wait, but what if a and b are the same? But the problem states they are different primes. So we don't have to consider the case where a and b are the same. So 2ab is the product of three distinct primes: 2, a, and b, each with exponent 1. So the number of divisors is (1+1)(1+1)(1+1) = 8.

Therefore, regardless of which primes a and b we choose (as long as they are different and greater than 2), the expression simplifies to 2ab, which has 8 divisors.

So the answer is 8. Therefore, the number of whole-number divisors is 8.

But wait, let me check if there's any other possible interpretation. The original expression: a(2a + b) - 2a² + ab. Let me verify once again.

Expanding a(2a + b) gives 2a² + ab. Then subtract 2a²: 2a² - 2a² = 0. Then add ab: 0 + ab = ab. Wait, is that correct? Wait, no. Wait, the original expression is a(2a + b) - 2a² + ab. So that's equal to [a(2a + b)] - 2a² + ab. So first compute a(2a + b) = 2a² + ab. Then subtract 2a²: 2a² + ab - 2a² = ab. Then add ab: ab + ab = 2ab. So yes, that's correct. So the expression simplifies to 2ab.

Therefore, the number of divisors is 8.

Therefore, the answer is 8. So I think that's the correct answer. Let me just recap to make sure I didn't miss anything.

1. Simplify the expression: 2ab.
2. Factorize into primes: 2 * a * b.
3. Since a and b are distinct primes greater than 2, the exponents are all 1.
4. Number of divisors is (1+1)(1+1)(1+1) = 8.
5. Verified with examples.
6. All steps checked out.

Therefore, the answer is 8.
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