Sum of Numbers satisfying the given conditions:
To find the sum of numbers between 500 and 700 that satisfy the conditions provided, we need to consider numbers that leave no remainder when divided by 6, 8, and 12, as well as numbers that leave a remainder of 5 in each case.

Numbers Divisible by 6, 8, and 12:
- To find numbers divisible by 6, 8, and 12, we need to find the least common multiple (LCM) of these numbers.
- LCM of 6, 8, and 12 is 24.
- Therefore, the numbers divisible by 6, 8, and 12 are multiples of 24 within the range of 500 to 700.

Numbers leaving a Remainder of 5:
- To find numbers that leave a remainder of 5 when divided by 6, 8, and 12, we need to add 5 to the multiples of 24.
- The numbers that satisfy this condition are 29, 53, 77, ..., 677.

Calculating the Sum:
- The numbers within the range of 500 to 700 that satisfy both conditions are 524, 548, 572, ..., 692.
- This is an arithmetic sequence with a common difference of 24.
- The sum of an arithmetic sequence can be calculated using the formula: \( \frac{n}{2} \times (first term + last term) \), where n is the number of terms.
- Calculating the number of terms in the sequence: \( \frac{last term - first term}{common difference} + 1 \)
- Substituting the values, we get the sum of the numbers satisfying the given conditions.
Therefore, by following the above steps, the sum of numbers between 500 and 700 that leave no remainder when divided by 6, 8, and 12, and leave a remainder of 5 in each case can be calculated.