Math101 · Independent Ontario learning library
Math101Greatest Common Factor
The greatest common factor (GCF) of two or more nonzero whole numbers is the largest positive integer dividing every number exactly.
The GCF forms the largest equal groups, simplifies fractions, and extracts the largest shared factor from algebraic terms. It is a compact description of common divisibility.
Intuition and core definition
The greatest common factor (GCF) of two or more nonzero whole numbers is the largest positive integer dividing every number exactly. For $18$ and $30$, the common positive factors include $1,2,3,6$, so the GCF is $6$. The GCF is also called the greatest common divisor (GCD).
Notation, language, and conditions
$\gcd(a,b)$ denotes the GCF; usually $\gcd(a,b)>0$ unless both inputs are zero, a case commonly left undefined in elementary courses. In prime factorizations, the GCF uses only primes present in every number and takes the smallest shared exponent.
Why this idea matters
The greatest common factor identifies the largest equal unit that can divide several whole quantities without leftovers.
A dependable method
- Prime-factor each number or use the Euclidean algorithm for larger inputs.
- Align identical prime bases.
- Keep only the primes common to all numbers.
- For each common prime, choose the smallest exponent.
- Multiply those factors and verify that every original number divides evenly by the result.
