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E3. A Fraction Sequence

Let S be the sequence of all proper fractions with denominator ≤ 20, arranged in order from smallest to largest, so the first fraction is 0/1 = 0 and the last one is 1/1 = 1. Represent each fraction in lowest terms. Define the gap between two consecutive fractions in this sequence to be the difference of the smaller from the larger.

  1. Determine the smallest gap in the sequence, and find two fractions that have this gap between them.

  2. The gap between the first two fractions is 1/20 – 0/1 = 1/20, as is the gap between the last two fractions. Not counting these fractions, find two fractions that have a gap that is as large as possible.

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