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Reduced fraction


An irreducible fraction (or fraction in lowest terms or reduced fraction) is a fraction in which the numerator and denominator are integers that have no other common divisors than 1 (and -1, when negative numbers are considered). In other words, a fraction ab is irreducible if and only if a and b are coprime, that is, if a and b have a greatest common divisor of 1. In higher mathematics, "irreducible fraction" may also refer to rational fractions such that the numerator and the denominator are coprime polynomials. Every positive rational number can be represented as an irreducible fraction in exactly one way.

An equivalent definition is sometimes useful: if a, b are integers, then the fraction ab is irreducible if and only if there is no other equal fraction cd such that |c| < |a| or |d| < |b|, where |a| means the absolute value of a. (Two fractions ab and cd are equal or equivalent if and only if ad = bc.)

For example, 14, 56, and −101100 are all irreducible fractions. On the other hand, 24 is reducible since it is equal in value to 12, and the numerator of 12 is less than the numerator of 24.

A fraction that is reducible can be reduced by dividing both the numerator and denominator by a common factor. It can be fully reduced to lowest terms if both are divided by their greatest common divisor. In order to find the greatest common divisor, the Euclidean algorithm or prime factorization may be used. The Euclidean algorithm is commonly preferred because it allows one to reduce fractions with numerators and denominators too large to be easily factored.


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