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A fraction is in lowest terms (also called "simplest form" or "reduced form") when its numerator and denominator share no common factor besides 1 — meaning there's no smaller equivalent fraction left to find. Dividing both the numerator and denominator by their greatest common divisor (GCD) gets there in a single step, rather than having to guess-and-check smaller factors one at a time. Reducing fractions to lowest terms is standard practice in nearly every context where a fraction is reported as a final answer: math homework and standardized tests almost always expect answers in simplest form, cooking and construction measurements are easier to work with in reduced form (3/4 cup rather than 12/16 cup), and probability and statistics results are conventionally simplified so the numbers convey the underlying relationship as clearly as possible.
How it works
Enter a numerator and denominator. The calculator finds their greatest common divisor (GCD) using the Euclidean algorithm — an efficient, ancient technique that repeatedly replaces the larger of the two numbers with the remainder of dividing it by the smaller one, continuing until a remainder of exactly 0 is reached; the last nonzero remainder produced along the way is the GCD. Once the GCD is known, dividing both the numerator and denominator by it in one step produces the fully reduced, lowest-terms fraction — there's no need to search for it through trial and error or repeated smaller simplifications.
- Enter numerator.
- Enter denominator.
- Click Calculate to see your results.
Examples
18/24
GCD(18, 24) = 6 (found via the Euclidean algorithm: 24 = 1×18 + 6, then 18 = 3×6 + 0, so 6 is the last nonzero remainder). Dividing both parts by 6: 18/24 = 3/4.
45/60
GCD(45, 60) = 15 (45 = 45, 60 = 45+15; 60 mod 45 = 15, then 45 mod 15 = 0, so 15 is the GCD). Dividing both parts by 15: 45/60 = 3/4.
17/34
GCD(17, 34) = 17, since 34 is exactly twice 17 (34 mod 17 = 0 immediately). Dividing both parts by 17: 17/34 = 1/2.
Common mistakes to avoid
- Dividing by a common factor that isn't the greatest one, leaving the fraction only partially reduced and requiring a second simplification pass.
- Forgetting to apply the Euclidean algorithm (or an equivalent check) and simply guessing at a GCD, which can miss a larger shared factor.
- Assuming a fraction is already in lowest terms just because a couple of small numbers (like 2 or 3) don't divide both parts evenly, without checking larger possible factors.
- Applying a simplification to only the numerator or only the denominator instead of both, which changes the fraction's actual value.