Math101 · Independent Ontario learning library
Math101learn.math101.caGreatest 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.
Worked example
Representations and interpretation
A Venn diagram of prime factors places shared copies in the overlap; their product is the GCF. In a grouping context, the GCF gives the largest equal group size that uses all items with none left over.
Reasoning about variations
For $24=2^3\cdot3$ and $40=2^3\cdot5$, all three copies of $2$ are shared, so the GCF is $8$. If numbers are relatively prime, the overlap is empty and their GCF is $1$.
Common mistakes
How to check your work
- Divide every input by the proposed GCF and require integer quotients.
- Check that the quotients have no common factor greater than $1$.
- Compare with the smaller input; a GCF cannot exceed it.
Practice
- Find $\gcd(48,72)$.
- Find the GCF of $18,30,42$.
- What is the GCF of $25$ and $18$?
Answers and brief solutions
Show answers
- $24$ $48=2^4\cdot3$ and $72=2^3\cdot3^2$ share $2^3\cdot3=24$.
- $6$ All three share $2\cdot3$, and no larger factor divides all three.
- $1$ They share no prime factor, so they are relatively prime.
Synthesis and transfer
Cutting two ribbon lengths into the longest identical pieces turns the GCF into a physical unit; multiplying the piece length by each piece count reconstructs both originals.
Suppose the ribbons measure $84$ cm and $126$ cm. Their prime factorizations share $2\cdot3\cdot7=42$, so the longest identical piece is $42$ cm; the cuts produce two and three pieces respectively. Choosing a smaller common factor such as $21$ would work physically but would not satisfy “longest.” The same extraction appears algebraically when $84x^3+126x^2$ becomes $42x^2(2x+3)$: minimum shared exponents determine the variable part. Expanding the product reconstructs the original terms, while checking the bracket's coefficients and variable powers for another common factor confirms that the greatest factor—not merely a common one—was removed.
Related topics
Teaching and accessibility note
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Find $\gcd(48,72)$.
- $48=2^4\cdot3$ and $72=2^3\cdot3^2$ share $2^3\cdot3=24$.
End of lesson
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