GCF Finder(GCD, HCF)

About GCF Finder(GCD, HCF)

The Greatest Common Factor(GCF), also known as the Greatest Common Divisor(GCD) or Highest Common Factor(HCF), refers to the largest divisor shared by two or more integers. The maximum common divisor of a and b is denoted as (a, b), and similarly, the maximum common divisor of a, b, and c is denoted as (a, b, c). The same notation applies to the maximum common divisor of multiple integers. There are various methods for finding the maximum common divisor, including prime factorization, Euclid's Algorithm, and so on.

If the number a can be divided by the number b, a is called a multiple of b, and b is called a divisor of a. Both divisors and multiples represent the relationship between one integer and another, and cannot exist alone. If we can say that 16 is a multiple of a certain number and 2 is a divisor of a certain number, we cannot say in isolation that 16 is a multiple and 2 is a divisor.

The common divisor among several integers is called the common divisor of these numbers; The largest one among them is called the Greatest Common Divisor(GCD) of these numbers. For example, the common divisors of 12 and 16 are 1, 2, and 4, with the largest being 4. 4 is the Greatest Common Divisor(GCD) of 12 and 16, usually denoted as GCF(12, 16)=4. The maximum common divisor of 12, 15 and 18 is 3, denoted as GCF(12, 15, 18)=3.

The main methods for finding the Greatest Common Factor(GCF) of two integers are:

1. Prime factorization: List the prime factorization formulas for two numbers separately and calculate the product of the common terms.
2. Euclid's Algorithm: dividing two numbers by their common prime factors until they are mutually prime, the product of all divisors is the Greatest Common Divisor(GCD).

1. The prime factorization

Representing a positive integer greater than 1 as the product of its prime factors is called factoring this positive number into prime factors. The common factor of several numbers is the factor of the Greatest Common Factor(GCF) of these numbers. From this and the definition of the Greatest Common Factor(GCF), we can obtain the factorization prime factor method for finding the Greatest Common Factor(GCF).

Steps:

1. Write the standard decomposition formula for each number;
2. Write the common prime factor and its minimum power for each factorization, and write the resulting power in the form of continuous multiplication.

For example, finding the Greatest Common Factor(GCF) of (60,108,24)

Step 1: Separate all factors of each number:

60=2²X3X5, 108=2²X3³, 24=2³X3

Step 2: Find the Greatest Common Factor(GCF): 2² X3

Therefore: GCF(60,108,24)=2²X3=12.

In practice, this method is only feasible when the number is relatively small, as performing prime factorization on larger numbers usually takes a lot of time.

2. Euclid's Algorithm:

Euclid's Algorithm refers to the maximum common divisor used to calculate two non-negative integers a and b. Calculate the formula gcd (a,b)=gcd (b, a mod b). Based on the following principle: the maximum common divisor of two integers is equal to the maximum common divisor of the smaller number and the remainder of the division between the two numbers. For example, if you want to find the maximum common divisor of 252 and 105, because 252 ÷ 105 = 2......42, then this maximum common divisor is also the maximum common divisor of 42 and 105. 105 ÷ 42=2......21, and so on until 105 ÷ 21=5......0, so GCF(252,105) = 21. During this process, the larger numbers are reduced, so continuing the same calculation can continuously reduce these two numbers until the remainder is zero. At this point, the remaining number that has not yet become zero is the Greatest Common Divisor(GCD) of two numbers.

For example, using Euclid's Algorithm to find the maximum common divisor of two positive integers, 1997 and 615.

1997 ÷ 615 = 3 (remaining 152)

615 ÷ 152 = 4 (remaining 7)

152 ÷ 7 = 21 (remaining 5)

7 ÷ 5 = 1 (2 remaining)

5 ÷ 2 = 2 (1 remaining)

2 ÷ 1 = 2 (remaining 0)

At this point, the Greatest Common Divisor(GCD) is 1.

Repeatedly performing division with divisor and remainder, when the remainder is 0, taking the current formula divisor as the Greatest Common Divisor(GCD) yields the Greatest Common Divisor(GCD) 1 for 1997 and 615.

Another example is using Euclid's Algorithm to find the maximum common divisor of 319 and 377, i.e. GCF(319,377).

319 ÷ 377 = 0 (remaining 319)

377 ÷ 319 = 1 (remaining 58)

319 ÷ 58 = 5 (remaining 29)

58 ÷ 29 = 2 (remaining 0)

(58, 29) = 29;

So GCF (319,377) = 29.

If we are calculating the maximum common divisor of three numbers, such as GCF (15,30,90), since GCF (x, y, z)=GCF (GCF (x, y), z), we first find GCF(90,30):

90 ÷ 30 = 3...... 0;

GCF (90,30) = 30;

Next, find GCF(30, 15):

30 ÷ 15=2...... 0;

GCF (30,15)=15;

Therefore, GCF (15,30,90)=15.

Prime factorization and Euclid's Algorithm are commonly used and relatively simple methods for finding the Greatest Common Factor(GCF).

In the GCF Finder we provide, we use the Factorization method because it is more intuitive and easier to help you understand the principles.

Using the GCF Finder is very simple. You only need to enter numbers in the input box, separated by commas. To calculate GCF. You need to enter at least 2 different positive integers (please do not enter negative values). When you repeatedly enter the same number, the duplicates will be removed. We support finding GCF of up to 10 positive integers.