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Greatest common factor

greatest common factor

One of the more useful techniques for simplifying expressions and numbers is finding the greatest common factor. The largest number (factor) that divides two or more numbers is called the greatest common factor or GCF.

Two methods are used to find the greatest common factor. The first method includes writing down all the factors of two or more numbers. After that, we need to write all the common factors for each number. The greatest common factor of the numbers is the largest number in that list.

common factors

First GCF method

Let’s try this method on the next example. We need to find the largest common factor for numbers 36 and 24.
First, we need to list all the factors for each number. In order to find all the factors of the number 36, we need to see if the number can be divided by a number between 1 and 36. If numbr 36 can be divided by that number without leaving a remainder, then that number is one of the factors of the number 36.
-> Number 36 can be divided by 1 so one of the factors is 1.
-> Number 36 can be divided by 2 so one of the factors is 2.
-> Number 36 can be divided by 3 so one of the factors is 3.
-> Number 36 can be divided by 4 so one of the factors is 4.
-> Number 36 can’t be divided by 5.
-> Number 36 can be divided by 6 so one of the factors is 6.
-> Number 36 can’t be divided by 7 and 8.
-> Number 36 can be divided by 9 so one of the factors is 9.
-> Number 36 can’t be divided by 10 and 11.
-> Number 36 can be divided by 12 so one of the factors is 12.
-> Number 36 can’t be divided by any number between 13 and 17.
-> Number 36 can be divided by 18 so one of the factors is 18.
-> Number 36 can’t be divided by any number between 19 and 35.
-> Number 36 can be divided by 36 and this is the last factor.

We need to list the factors. They are 1, 2, 3, 4, 6, 9, 18 and 36.

We need to repeat the process for number 24:
-> Number 24 can be divided by 1 so one of the factors is 1.
-> Number 24 can be divided by 2 so one of the factors is 2.
-> Number 24 can be divided by 3 so one of the factors is 3.
-> Number 24 can be divided by 4 so one of the factors is 4.
-> Number 24 can’t be divided by 5.
-> Number 24 can be divided by 6 so one of the factors is 6.
-> Number 24 can’t be divided by 7.
-> Number 24 can be divided by 8 so one of the factors is 8.
-> Number 24 can’t be divided by 9, 10 or 11.
-> Number 24 can be divided by 12 so one of the factors is 12.
-> Number 24 can’t be divided by any number between 13 and 23.
-> Number 24 can be divided by 24 and this is the last factor.
Now we need to list these factors. They are: 1, 2, 3, 4, 6, 8, 12 and 24.
Let’s compare them to the factors of 36: 1, 2, 3, 4, 6, 9, 12, 18 and 36.
Now, we need to list the common factors for these numbers and pick the highest one: 1, 2, 3, 4, 6, 12.
We can see that the greatest common factor is of 24 and 36 is 12.

gcf of 24 and 36

Second GCF method

The second method for finding the greatest common factor is to list all the prime factors for the two numbers. After that, we need to multiply the common prime numbers and we will get the greatest common factor.
Let’s try this method out with the same numbers from the previous example.

First we need to list the prime factors of 36.

prime factors of 24

To find the prime factors, we need to start dividing 36 with the lowest possible number that can divide 36.

We can start with number 2. Number 2 can divide number 36 and the result is 18. That means that number 2 is the first prime factor.
-> Number 2 can divide 18 and the result is 9. That means that the second prime factor is 2.
-> Number 2 can’t divide 9. The next number that we should try is number 3. Number 3 can divide number 9. The result is number 3, so 3 is the third prime factor.
-> Number 3 can divide 3. The result is 1. The fourth prime factor is number 3.
-> Since the last result is 1, that means that the calculation of the prime factors is done.
-> The prime factors of number 36 are:

Using the same method, we can find the prime factors for number 24.

prime factor of 36

The prime factors of 24 are: 2 \cdot 2 \cdot 2 \cdot 3
The prime factors of number 36 are: \ 2 \cdot 2 \cdot 3 \cdot 3 = 36
Now we need to pick the common prime factors. We can see that the common prime factors are 2,2 and 3. When we multiply 2 \cdot 2 \cdot 3 we get 12 which is the greatest common factor.

 

Greatest common factor worksheets

  GCF of two numbers up to 30 (92.3 KiB, 686 hits)

  GCF of two numbers up to 50 (97.8 KiB, 799 hits)

  GCF of two numbers up to 100 (112.0 KiB, 572 hits)

  GCF of two numbers up to 500 (146.7 KiB, 802 hits)

  GCF of two numbers up to 1000 (165.3 KiB, 1,785 hits)

  GCF of three numbers up to 30 (79.4 KiB, 618 hits)

  GCF of three numbers up to 50 (101.6 KiB, 451 hits)

  GCF of three numbers up to 100 (109.0 KiB, 1,786 hits)

  GCF of three numbers up to 500 (135.7 KiB, 1,574 hits)

  GCF of three numbers up to 1000 (189.3 KiB, 1,646 hits)

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