GCF Калькулятор
Найти the Greatest Common Factor of two or more numbers using prime factorization or the Euclidean algorithm.
Find the Greatest Common Factor
Venn Diagram (Common Prime Factors)
GCF is the intersection of prime factors
Math Keypad
Understanding the Greatest Common Factor (GCF)
The Greatest Common Factor (GCF) of two or more numbers is the largest positive integer that divides all of them without leaving a remainder. It is also referred to as the Greatest Common Divisor (GCD) or Highest Common Factor (HCF).
Two Ways to Find the GCF
Method 1
Prime Factorization
Best for smaller numbers. Break each number down into its prime factors, then multiply the common primes.
- Write the prime factorization of each number.
- Identify common prime factors.
- Choose the lowest exponent of each.
- Multiply them together.
Method 2
Euclidean Algorithm
Best for large numbers. Repeatedly divide the larger number by the smaller and take the remainder.
- Divide the larger by the smaller and find the remainder.
- Divide the previous divisor by the remainder.
- The last divisor when remainder is 0 is the GCF.
Relationship Between GCF and LCM
For any two positive integers a and b, their GCF and Least Common Multiple (LCM) are mathematically linked by their product:
GCF & LCM Formula
GCF(a, b) × LCM(a, b) = a × b
Common GCF & LCM Reference
| Numbers | Common Factors | GCF | LCM |
|---|---|---|---|
| 12, 18 | 1, 2, 3, 6 | 6 | 36 |
| 24, 36 | 1, 2, 3, 4, 6, 8, 12 | 12 | 72 |
| 15, 35 | 1, 5 | 5 | 105 |
Practical Applications
- • Simplifying fractions efficiently by dividing the numerator and denominator by their GCF.
- • Distributing quantities of different items into the largest possible equal groups.
Tips & Important Notes
- • If the GCF of two numbers is 1, they are called coprime or relatively prime numbers.
- • The GCF of any prime numbers is always 1 unless one is a multiple of the other.
Frequently Asked Questions
What is the Greatest Common Factor (GCF)?
The Greatest Common Factor (GCF), also known as the Greatest Common Divisor (GCD) or Highest Common Factor (HCF), is the largest positive integer that divides each of the given numbers without leaving a remainder.
How do I find the GCF of two numbers?
You can find the GCF using two main methods: 1) Prime factorization — break each number into prime factors and multiply the common ones. 2) Euclidean algorithm — repeatedly divide the larger number by the smaller and take the remainder until you reach 0.