Guide

How to Simplify Fractions Using the Greatest Common Divisor

A fraction is in simplest form when its numerator and denominator share no common divisor greater than 1, and the denominator is positive. 24/36 is not simplest. 2/3 is. The process is: find the greatest common divisor, divide both parts by it, and fix the sign. The simplify fractions tool uses BigInt so large integers stay exact.

What simplest form means

Simplest form is a canonical name for a rational number, not a change of value. 24/36 and 2/3 mark the same point on the number line. Writing 2/3 makes comparisons and later arithmetic easier. Integers are already simplest: 4/1 is stored as 4. Zero is 0/1, not 0/8.

Common factors

A common factor of 24 and 36 is any positive integer that divides both. 2, 3, 4, 6, and 12 all divide both. Dividing by 2 gives 12/18, which is smaller but still reducible. You must divide by the greatest common factor, 12, to finish in one step.

Greatest common divisor

The GCD of two integers is the largest positive integer that divides both. gcd(24, 36) = 12. gcd(7, 13) = 1. gcd(0, 8) = 8, which is why 0/8 reduces to 0/1. The GCD of a number and 1 is 1, so n/1 is already reduced as an integer.

Euclidean algorithm

The Euclidean algorithm computes GCD without listing factors. Replace the larger number by the remainder when it is divided by the smaller, and repeat until the remainder is 0. The last nonzero remainder is the GCD. For 36 and 24: 36 = 1 × 24 + 12. 24 = 2 × 12 + 0. GCD 12. Signs are ignored by taking absolute values first, so gcd(−4, 8) = 4. This site implements that loop with BigInt, not with floating-point remainders.

Reducing numerators and denominators

Simplified numerator = numerator / GCD. Simplified denominator = denominator / GCD. 24/12 = 2. 36/12 = 3. Result 2/3. If the denominator was negative, multiply both by −1 so the denominator becomes positive. 4/−8 has GCD 4 and becomes −1/2 after the sign move.

Worked example: 24/36

gcd(24, 36) = 12. 24/12 = 2. 36/12 = 3. 2/3. Confirm with the simplifier. A second example: 10/20. GCD 10. Result 1/2.

Equivalent fractions

Every unreduced fraction is equivalent to its reduced form. 24/36, 12/18, and 2/3 are equivalent. When you compare 1/2 and 2/4, cross-multiplication shows equality: 1 × 4 = 2 × 2. Simplifying first also shows both are 1/2. See the comparison calculator.

Negative fractions

−4/8 reduces to −1/2. 4/−8 also becomes −1/2. −4/−8 becomes 1/2. The value’s sign lives on the numerator. If you only divide by 4 and leave 1/−2, the site still canonicalizes to −1/2 before display.

Zero numerator

0/8 is zero. After reduction it is 0/1, displayed as 0. 0/0 is invalid because the denominator is zero. 5/0 is invalid for the same reason. Simplifying cannot repair a zero denominator.

Already simplified fractions

7/13 has GCD 1. The tool reports: “This fraction is already in simplest form.” It does not invent a reduction. 2/3 is the same. If someone writes 2/3 and asks to simplify, the honest answer is that no further cancellation exists.

Checking the result

Multiply the simplified numerator and denominator by the GCD and recover the original magnitudes, then restore sign. 2 × 12 = 24. 3 × 12 = 36. For 7/13, GCD 1 recovers 7/13 immediately. Avoid checking only with a calculator’s decimal: 2/3 and 0.67 are not the same claim. Related: adding fractions always simplifies the sum, and the formula page restates the GCD rule.