First solution (shortcut):

If 700 was included in our set of tree-digit integers, the number of integers that satisfied problem's condition for 700-799 set would be equal to the number for 800-899 set as well as 900-999 set. In other words, the total number would be divisible by 3 (we have 3 equal sets). Since we should exclude 700 from consideration, the right answer should be divisible by 3 with a remainder of 2.

Now, let's plug the numbers:

A) 90 is divisible by 3. It can't be a right answer;

B) 82 is divisible by 3 with remainder of 1. It can't be a right answer;

C) 80 is divisible by 3 with remainder of 2. It can be a right answer;

D) 45 is divisible by 3. It can't be a right answer;

E) 36 is divisible by 3. It can't be a right answer;

Only C is divisible by 3 with a remainder of 2 and is the right answer.