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Let represent the smaller of the two consecutive odd integers. Then must represent the next consecutive odd integer.
The reciprocal of the smaller is then , and the reciprocal of the larger one is . Then the difference between the reciprocals is:
Combine the fractions in the LHS using the common denominator
Collect terms on the left:
Now we have two equal fractions with equal numerators -- therefore the denominators must be equal as well, so:
Solve the factorable quadratic. Discard the negative root. The positive root will be the smaller of the two positive consecutive odd integers.
