What is GCF of 24 and 36?
GCF(24, 36) = 12. Twelve is the largest integer dividing both 24 and 36 without remainder.
Greatest common factor
GCF = 6
Formula: GCF(a, b) = GCF(b, a mod b)
12: 1, 2, 3, 4, 6, 12
18: 1, 2, 3, 6, 9, 18
Find the largest integer that divides every input without remainder—simplify fractions, split materials evenly, and trace Euclidean algorithm steps.
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The GCF of two or more integers is the largest whole number that divides every input without a remainder. GCF(24, 36) = 12, so 24/36 simplifies to 2/3 when you divide both by 12.
Greatest common factor
GCF = 6
Formula: GCF(a, b) = GCF(b, a mod b)
12: 1, 2, 3, 4, 6, 12
18: 1, 2, 3, 6, 9, 18
The greatest common factor—also called the greatest common divisor (GCD)—answers one practical question: what is the biggest chunk you can carve out of every number at once?
If you have 24 pencils and 36 erasers to pack into identical gift bags with nothing left over, the bag size is limited by shared factors. Factors of 24 include 1, 2, 3, 4, 6, 8, 12, and 24. Factors of 36 include 1, 2, 3, 4, 6, 9, 12, 18, and 36. The largest number on both lists is **12**, so GCF(24, 36) = 12.
Mathematically, the GCF is the product of prime factors raised to their **minimum** exponents across all inputs. That is why prime factorization and the Euclidean algorithm both land on the same answer—the algorithm is simply faster for large integers.
**GCF vs LCM:** GCF captures overlap (what is shared); LCM captures coverage (what is needed to hit every number). For two positive integers, GCF(a,b) × LCM(a,b) = |a×b|. Use the LCM calculator when you need a common denominator, not a shared divisor.
GCF finds the largest shared divisor—divide every input by it to simplify fractions or split items into equal largest groups.
Concise answers for common searches — definitions, steps, and comparisons.
GCF(24, 36) = 12. Twelve is the largest integer dividing both 24 and 36 without remainder.
Divide numerator and denominator by their GCF. Example: 18/24 → GCF is 6 → 3/4.
Replace (a,b) with (b, a mod b) until the remainder is 0. The last non-zero divisor is the GCF.
Yes. Greatest common factor and greatest common divisor name the same largest shared whole-number divisor.
This calculator accepts two or more positive integers, runs the Euclidean algorithm pairwise, and optionally shows prime-factor steps for transparency.
Formula
GCF(a, b) via Euclidean algorithm: repeat GCF(b, a mod b) until b = 0; answer is a. For lists: GCF(a, b, c) = GCF(GCF(a, b), c). Prime method: GCF = ∏ p_i^min(e_i across inputs).Type positive whole numbers separated by commas or spaces—e.g. 48, 180, 300. The tool processes them as a set sharing one GCF.
The largest shared divisor appears with optional steps showing Euclidean remainders or prime overlap.
Divide each fraction's numerator and denominator by the GCF of those two parts to reach lowest terms.
If the task asks for a common denominator instead, open the LCM calculator—GCF and LCM solve opposite but linked problems.
Input
12 and 18Output
GCF = 6Shared factors: 1, 2, 3, 6. Largest is 6. Fraction 12/18 → 2/3.
Input
8 and 15Output
GCF = 1No shared prime factors—numbers are relatively prime. 8/15 is already lowest terms.
Input
24, 36, 60Output
GCF = 12GCF(24,36)=12; GCF(12,60)=12. All three share 12 but not 24.
Input
1071 and 462Output
GCF = 211071 = 2×462 + 147; 462 = 3×147 + 21; 147 = 7×21 + 0 → GCF 21.
Input
360 and 504Output
GCF = 72360=2³×3²×5; 504=2³×3²×7. Shared primes at min powers: 2³×3²=72.
Input
17 and 51Output
GCF = 1751 = 3×17, so the smaller number is the GCF when it divides the larger.
Common real-world scenarios where this tool saves time.
Before adding 5/15 and 7/21, notice GCF(5,15)=5 and GCF(7,21)=7. Reduced forms 1/3 and 1/3 make common-denominator work cleaner.
Two boards measure 84 in and 126 in. GCF(84, 126) = 42, so the longest equal segment that fits both whole numbers of times is 42 inches (2 pieces and 3 pieces).
A room repeat is 48 cm by 72 cm. GCF(48, 72) = 24 cm is the largest square module that tiles both dimensions evenly.
Trace GCF(1071, 462) = 21 by hand, then confirm the remainder chain matches the tool output.
Step-by-step chains that connect related tools for common tasks.
These two tools answer opposite questions about the same pair of numbers.
| Measure | GCF | LCM | Typical use |
|---|---|---|---|
| Definition | Largest shared divisor | Smallest shared multiple | — |
| Example (12, 18) | 6 | 36 | Simplify vs add fractions |
| Prime exponents | Minimum power per prime | Maximum power per prime | Factor method |
| Product identity (two nums) | GCF × LCM = |a×b| | Same identity | Cross-check answers |
Use LCM Calculator when denominators must match.
| Inputs | GCF | Quick reason |
|---|---|---|
| 14, 49 | 7 | Both multiples of 7 |
| 27, 64 | 1 | No shared primes |
| 100, 250 | 50 | 100 divides 250 twice |
| 45, 60, 75 | 15 | All divisible by 15 |
For homework under 100, sketch divisibility by 2, 3, 5, and 11 before reaching for the tool—builds number sense.
Compute GCF(a,b), then GCF(result, c). The calculator does this automatically; mimic the order on paper.
Listing primes clarifies why exponents use the minimum power—helpful when introducing the prime number calculator.
After reducing a/b, multiply back—if GCF was correct, no smaller integer divides both parts.
GCF is the largest **divisor** shared by all inputs. LCM is the smallest **multiple** hit by every input. They solve different fraction tasks.
2 might divide both 24 and 36, but GCF is 12. List all common factors or use the Euclidean algorithm to avoid stopping early.
GCF reduces fractions. Adding unlike denominators needs LCM for a common denominator, not GCF of denominators alone.
Relatively prime numbers have GCF 1—that is correct, not an error.
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No. The GCF never exceeds the smallest number in the set. If one number divides another, the GCF equals the smaller number.
By convention GCF(0, n) = |n| for n ≠ 0. Zero shares every factor, so the magnitude of the non-zero input is used.
GCF(42, 56) = 14. Divide both parts: 42÷14 = 3 and 56÷14 = 4, giving 3/4 in lowest terms.
No. GCF(18, 24) = GCF(24, 18). The set of inputs is unordered.
Prime factorization lists each input as primes with exponents. GCF uses the lowest exponent for each prime present in all inputs.
Any common divisor of a and b also divides the remainder a mod b. The last non-zero remainder is the greatest such divisor.
Yes. GCF(a,b,c) = GCF(GCF(a,b), c). Extend pairwise for longer lists—the calculator accepts multiple integers.
GCF Calculator runs in your browser—integers you enter are not uploaded to EverydayTools servers.
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Reviewed by EverydayTools Editorial Team on 2026-07-03.
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