Can we prove the following is always an integer?

$$6b\sum\_{k=1}^bk\left\{\frac{ka}{b}\right\}$$

where $\{x\}=x-\lfloor x\rfloor$ denotes the fractional part operator.

UPDATE:

Through the calculation of several identities and going through heavy casework, I have proven the above formula; the proof, however, is too lengthy and messy to post here.