Modular Arithmetic
Unit fractions play an important role in modular arithmetic, as they may be used to reduce modular division to the calculation of greatest common divisors. Specifically, suppose that we wish to perform divisions by a value x, modulo y. In order for division by x to be well defined modulo y, x and y must be relatively prime. Then, by using the extended Euclidean algorithm for greatest common divisors we may find a and b such that
from which it follows that
or equivalently
Thus, to divide by x (modulo y) we need merely instead multiply by a.
Read more about this topic: Unit Fraction
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—Ralph Waldo Emerson (1803–1882)