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Two infinite products AAA and BBB are defined as follows:
A=∏n=2∞(1−1n3),B=∏n=1∞(1+1n(n+1)). A = \displaystyle\prod\limits_{n=2}^{\infty} \left(1-\frac{1}{n^3}\right), \quad B =\displaystyle\prod\limits_{n=1}^{\infty}\left(1+\frac{1}{n(n+1)}\right). A=n=2∏∞(1−n31),B=n=1∏∞(1+n(n+1)1).
If AB=mk,\frac{A}{B} = \frac{m}{k},BA=km, where mmm and kkk are relatively prime positive integers, determine 100m+k100m+k100m+k.
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