Maximum of e

Level pending

Let \( a, b, c, d, e \) be positive integers which satisfies \[ a < b < c < d < e \] and \[ \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} + \frac{1}{e} = 1 \] What are the last three digits of the maximum value of \( e\) ?

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