Introduction
To solve the given problem, we need to find the number of integers between 245 and 720 that are divisible by 12 or 15 but not by 8.

Formulae Used
To solve this problem, we need to use the following formulae:
- A number is divisible by 12 if it is divisible by both 3 and 4.
- A number is divisible by 15 if it is divisible by both 3 and 5.
- A number is not divisible by 8 if its last three digits are not divisible by 8.

Solution
To find the required numbers, we need to follow these steps:
1. Find the number of integers between 245 and 720 that are divisible by 12.
2. Find the number of integers between 245 and 720 that are divisible by 15.
3. Find the number of integers between 245 and 720 that are divisible by both 12 and 15.
4. Subtract the number of integers divisible by both 12 and 15 from the sum of the numbers divisible by 12 and the numbers divisible by 15.
5. Find the number of integers between 245 and 720 whose last three digits are not divisible by 8.
6. Subtract the number of integers whose last three digits are divisible by 8 from the result obtained in step 4.

Step 1: Find the number of integers between 245 and 720 that are divisible by 12.
To find the number of integers between 245 and 720 that are divisible by 12, we need to find the first and the last multiples of 12 in this range.

The first multiple of 12 in this range is 252 (12 x 21).
The last multiple of 12 in this range is 708 (12 x 59).

Therefore, there are 39 integers between 245 and 720 that are divisible by 12.

Step 2: Find the number of integers between 245 and 720 that are divisible by 15.
To find the number of integers between 245 and 720 that are divisible by 15, we need to find the first and the last multiples of 15 in this range.

The first multiple of 15 in this range is 255 (15 x 17).
The last multiple of 15 in this range is 720 (15 x 48).

Therefore, there are 32 integers between 245 and 720 that are divisible by 15.

Step 3: Find the number of integers between 245 and 720 that are divisible by both 12 and 15.
To find the number of integers between 245 and 720 that are divisible by both 12 and 15, we need to find the first and the last multiples of the LCM of 12 and 15, which is 60.

The first multiple of 60 in this range is 300 (60 x 5).
The last multiple of 60 in this range is 660 (60 x 11).

Therefore, there are 9 integers between 245 and 720 that are divisible by both 12 and 15.

Step 4: Subtract the number of integers divisible by both 12 and 15 from the sum of the numbers divisible by 12 and the numbers divisible by 15