$$A$$ and $$B$$ are two different numbers selected from the first forty natural numbers. The largest value that $$\frac{A \cdot B}{A - B}$$ can have is __________

Correct Answer: 1560

Solution

For $$A > B$$, $$\frac{AB}{A - B} = \frac{1}{\frac{1}{B} - \frac{1}{A}}$$, which is largest when $$\frac{1}{B} - \frac{1}{A}$$ is smallest. That happens for the largest consecutive pair, $$A = 40$$ and $$B = 39$$. The value is $$\frac{40 \times 39}{1} = 1560$$.

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