Let $N$ be the number of ways of distributing $52$ identical balls into $4$ distinguishable boxes such that no box is empty and the difference between the number of balls in any two of the boxes is not a multiple of $6$ If $N=100a+b$, where $a,b$ are positive integers less than $100$, find $a+b.$
1. Let the number of balls in each box be \(a_1, a_2, a_3,\) and \(a_4\). We need to distribute 52 balls such that no box is empty and the difference between the number of balls in any two boxes is not a multiple of 6.
2. Notice that the set \(\{a_1 \pmod 6, a_2 \pmod 6, a_3 \pmod 6, a_4 \pmod 6\}\) can only hold three values: \(\{0,1,4,5\}, \{0,2,3,5\},\) and \(\{1,2,3,4\}\) since \(52 \equiv 4 \pmod 6\).
3. In all of these cases, the sum of the remainders is 10, which means that 42 balls remain. These 42 balls must be split into the boxes such that each box receives a number of balls that is a multiple of 6. This is equivalent to distributing 7 balls with no constraints.
4. The number of ways to distribute 7 balls into 4 boxes with no constraints is given by the stars and bars method: \(\binom{7+3}{3} = \binom{10}{3} = 120\).
5. In the case \(\{1,2,3,4\}\), there are \(4! = 24\) ways to distribute the remaining 10 balls, and all of these cases work because no box is empty.
6. In the other two cases, it's possible for one of the boxes to be empty. This depends on how the original 42 balls were distributed.
7. There are \(\binom{6}{3} = 20\) ways such that no boxes were empty after the first 42 balls were distributed. Then the remaining 10 balls have \(2 \cdot 4! = 48\) ways to be distributed based on either of the first 2 cases.
8. There are 4 ways to choose 1 box to be empty, and then \(\binom{6}{2} = 15\) ways to distribute the 42 balls in the other 3 boxes. In this case, the "0" in the set can't be distributed to the empty box, so there are only \(2 \cdot 3 \cdot 3! = 36\) ways.
9. There are \(\binom{4}{2} = 6\) ways to choose 2 boxes to be empty and then \(\binom{6}{1} \cdot 2 \cdot 2 \cdot 3! = 144\) ways to do the rest, following the previous procedure.
10. Finally, there are 4 ways to choose 3 boxes to be empty and only 12 ways to do the rest.

Thus, the total number of ways is:
\[
\binom{10}{3} \cdot 24 + \binom{6}{3} \cdot 48 + 4 \cdot \binom{6}{2} \cdot 36 + 6 \cdot \binom{6}{1} \cdot 24 + 4 \cdot 12 = 6912
\]

Given \(N = 100a + b\), we have \(N = 6912\). Therefore, \(a = 69\) and \(b = 12\).

The final answer is \( \boxed{ a + b = 69 + 12 = 81 } \).