Continued Fractions of Reciprocals
The continued fraction representations of a positive rational number and its reciprocal are identical except for a shift one place left or right depending on whether the number is less than or greater than one respectively. In other words, the numbers represented by and are reciprocals. This is because if is an integer then if then and and if then and with the last number that generates the remainder of the continued fraction being the same for both and its reciprocal.
For example,
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