Define the infinite products \[ A = \prod\limits_{i=2}^{\infty} \left(1-\frac{1}{n^3}\right) \text{ and }  B = \prod\limits_{i=1}^{\infty}\left(1+\frac{1}{n(n+1)}\right). \] If $\tfrac{A}{B} = \tfrac{m}{n}$ where $m,n$ are relatively prime positive integers, determine $100m+n$.

[i]Proposed by Lewis Chen[/i]