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basic explanation:

The greatest common factor is the largest common factor of one or more numbers. The most common ways to find the greatest common factor are factoring, prime factorization, and Euclid's algorithm. Our calculator for finding the greatest common factor offers an option to show work using all three of these methods. They are explained in more detail in the in-depth explanation section below. GCF is a common shorthand that stands for greatest common factor.

in-depth explanation:

factoring

One method for finding the GCF of one or more numbers is by factoring. To factor a number, begin by dividing the number being factored by 1, and see if there is a remainder of 0. If there is, then the number isn't a factor. If the remainder isn't 0, the number is a factor. Since 1 is a factor of every number, there will never be a remainder of 0 on this step. Next, however, you must increase the divisor by 1, which results in you dividing the number being factor by 2. As before, if there is a remainder of 0, the number isn't a factor. Otherwise, the number is a factor. Continue this process until the divisor is greater than the number being factored. All the numbers you got that had a remained that wasn't 0 is a factor. To find the GCF, you must factor every number you are trying to find the GCF for. Then, find the common factors. The common factors are factors of every number that was factored. Once you have the common factors, find the largest number among them. This is the greatest common factor.

prime factorization:

Another method for finding the GCF of two or more numbers is through prime factorization. Prime factorization is the process of finding prime factors that multiply to equal the number being prime factorized. There can be multiple occurrences of one prime factor within a single prime factorization. For example, the prime factorization of 12 is 2 × 2 × 3. To find the prime factorization of a number, first decide whether the number is itself a factor. If it is, the prime factorization of the number is simply the number itself. Otherwise, you must divide the number being prime factorized by the closest prime number less than the number itself. If the remainder is 0, the divisor is part of the prime factorization of the number being prime factorized. Otherwise, the divisor is not part of the prime factorization of the number being prime factorized. Continue dividing the number being factorized by a prime number each time, going to the closest prime number below the most recent prime number, until you are out of prime numbers. When you hit this point, you have found the prime factorization. To find the GCF using prime factorization, you must find the prime factorization for every number that is being used in the GCF problem. Once you have finished with that, you must find the occurrences of the common prime factors, which is a number that occurred in every prime factorization atleast once. Note that there can be multiple occurrences of common prime factors. Afterwards, multiply the occurrences of common prime factors together to find the GCF. If there are no common prime factors, the GCF is 1.

Euclid's algorithm:

The final method for finding the GCF of two or more numbers that we will explain is Euclid's algorithm. If you are only finding the GCF for more than two numbers with this technique, you must find the GCF for two numbers, then find the GCF of the answer to the previous and a third number, and so on. Otherwise, you only need to perform the following steps once. If at any moment you have a result of 0 in the following steps, move on to the next paragraph without performing the steps after the point where you reached 0. To find the GCF, first, subtract the smaller number from the larger number as many times as possible without getting a negative number. Afterwards, repeat the process, this time subtracting the result from the original smaller number. If you still haven't reached 0, subtract the preivous result by that step's result. Repeat this until you reach 0.

Once you reach 0, the GCF of the two numbers is the next-to-last small number result. If you are finding the GCF of more than two numbers, remember to go back and find the GCF of the answer and the next entered value if you have not already done so.

practice problems:

To fully understand this concept, it is important to utilize your knowledge through problems. Rather than creating your own, you can simply use some of the ones from our collection of GCF problems below. They are categorized into three different categories: easy, medium, and hard. It is recommended you begin with the easy problems and work your way up from there. If you are having difficulties solving any of the problems, you may use our GCF calculator which will calculate the answer and provide steps on how the problem was solved. You may also use the calculator to check you got the correct answer.

easy:

1.) The GCF of 2 and 4 is ?.

2.) The GCF of 3 and 9 is ?.

3.) The GCF of 12 and 40 is ?.

4.) The GCF of 7 and 32 is ?.

5.) The GCF of 12 and 35 is ?.

6.) The GCF of 31 and 16 is ?.

7.) The GCF of 39 and 38 is ?.

8.) The GCF of 84 and 23 is ?.

9.) The GCF of 94 and 58 is ?.

10.) The GCF of 48 and 67 is ?.

medium:

1.) The GCF of 132 and 494 is ?.

2.) The GCF of 394 and 232 is ?.

3.) The GCF of 623 and 212 is ?.

4.) The GCF of 233 and 844 is ?.

5.) The GCF of 245 and 744 is ?.

6.) The GCF of 456 and 123 is ?.

7.) The GCF of 383 and 232 is ?.

8.) The GCF of 1,320 and 84 is ?.

9.) The GCF of 232 and 833 is ?.

10.) The GCF lf 682 and 823 is ?.

hard:

1.) The GCF of 1,585 and 283 is ?.

2.) The GCF of 4,453 and 1,086 is ?.

3.) The GCF of 6,324 and 23,433 is ?.

4.) The GCF of 2,435 and 83,234 is ?.

5.) The GCF of 12,324 and 94,234 is ?.

6.) The GCF of 113,344 and 43,666 is ?.

7.) The GCF of 343,908 and 234,456 is ?.

8.) The GCF of 988,324 and 343,432 is ?.

9.) The GCF of 233,432 and 904,484 is ?.

10.) The GCF of 10,343,678 and 32,664,234 is ?.