 ## How to find the Greatest Common Factor of 19 and 31?

There are many methods we can apply to calculate the GCF of 19 and 31.

In our first method, we'll find out the prime factorisation of the 19 and 31 numbers.

In our second method, we'll create a list of all the factors of the 19 and 31 numbers.

These are the numbers that divide the 19 and 31 numbers without a remainder.

Once we have these, all we have to do is to find the one that is the biggest common number from the 2 lists.

Now let's look at each methods, and calculate the GCF of 19 and 31.

### Method 1 - Prime Factorisation

With the prime factorisation method, all we have to do is to find the common prime factors of 19 and 31, and then multiply them. Really simple:

#### Step 1: Let's create a list of all the prime factors of 19 and 31:

##### Prime factors of 19:

As you can see below, the prime factors of 19 are 19.

Let's illustrate the prime factorization of 19 in exponential form:

19 = 191

##### Prime factors of 31:

As you can see below, the prime factors of 31 are 31.

Let's illustrate the prime factorization of 31 in exponential form:

31 = 311

#### Step 2: Write down a list of all the common prime factors of 19 and 31:

As seen in the boxes above, the common prime factors of 19 and 31 are 1.

#### Step 3: All we have to do now is to multiply these common prime factors:

Find the product of all common prime factors by multiplying them:

11=1

##### Done!

According to our calculations above, the Greatest Common Factor of 19 and 31 is 1

### Method 2 - List of Factors

With this simple method, we'll need to find all the factors of 19 and 31, factors are numbers that divide the another number without a remainder, and simply identify the common ones, then choose which is the largest one.

#### Step 1: Create a list of all the numbers that divide 19 and 31 without a remainder:

List of factors that divide 19 without a remainder are:

1 and 19.

List of factors that divide 31 without a remainder are:

1 and 31.

#### Step 2: Identify the largest common number from the 2 lists above:

As you can see in the lists of factors from above, for the numbers 19 and 31, we have highlighted the number 1, which means that we have found the Greatest Common Factor, or GCF.

According to our calculations above, the Greatest Common Factor of 19 and 31 is 1

### Method 3 - Euclidean algorithm

The Euclidean algorithm says that if number k is the GCM of 19 and 31, then the number k is also the GCM of the division remainder of the numbers 19 and 31.

We follow this procedure until the reminder is 0.

The Greatest Common Divisor is the last nonzero number.

19, 31

#### Step 2

Take out, from the set, the smallers number as you divisor: 19

The remaining set is: 31

Find the reminder of the division between the number and the divisor

31 mod 19 = 12

Gather the divisor and all of the remainders and sort them in ascending order. Remove any duplicates and 0. Our set is:

12, 19

Repeat the process until there is only one number in the set.

Take out, from the set, the smallers number as you divisor: 12

The remaining set is: 19

Find the reminder of the division between the number and the divisor

19 mod 12 = 7

Gather the divisor and all of the remainders and sort them in ascending order. Remove any duplicates and 0. Our set is:

7, 12

Repeat the process until there is only one number in the set.

Take out, from the set, the smallers number as you divisor: 7

The remaining set is: 12

Find the reminder of the division between the number and the divisor

12 mod 7 = 5

Gather the divisor and all of the remainders and sort them in ascending order. Remove any duplicates and 0. Our set is:

5, 7

Repeat the process until there is only one number in the set.

Take out, from the set, the smallers number as you divisor: 5

The remaining set is: 7

Find the reminder of the division between the number and the divisor

7 mod 5 = 2

Gather the divisor and all of the remainders and sort them in ascending order. Remove any duplicates and 0. Our set is:

2, 5

Repeat the process until there is only one number in the set.

Take out, from the set, the smallers number as you divisor: 2

The remaining set is: 5

Find the reminder of the division between the number and the divisor

5 mod 2 = 1

Gather the divisor and all of the remainders and sort them in ascending order. Remove any duplicates and 0. Our set is:

1, 2

Repeat the process until there is only one number in the set.

Take out, from the set, the smallers number as you divisor: 1

The remaining set is: 2

Find the reminder of the division between the number and the divisor

2 mod 1 = 0

Gather the divisor and all of the remainders and sort them in ascending order. Remove any duplicates and 0. Our set is:

1

#### Step 3: Take the remaining number from our set

The Greatest Common Factor of 19 and 31 is 1

### Method 4 - Binary Greatest Common Divisor algorithm

The binary GCD algorithm, also known as Stein's algorithm or the binary Euclidean algorithm, is an algorithm that computes the greatest common divisor of two nonnegative integers. Stein's algorithm uses simpler arithmetic operations than the conventional Euclidean algorithm; it replaces division with arithmetic shifts, comparisons, and subtraction.

Although the algorithm in its contemporary form was first published by the Israeli physicist and programmer Josef Stein in 1967, it may have been known by the 2nd century BCE, in ancient China.

The list: 19, 31

#### Step 2: Pick the first number, 19. Subtract 19 from the remaining value(s) and divide the outcome by 2. Remove the duplicates and sort:

(31-19)/2 = 6

The resulting list: 6, 19

#### Step 3: Divide all of the remaining even values by 2, remove the duplicates and sort. Repeat the process if there are even numbers in the list:

6/2 = 3

The resulting list: 3, 19

#### Step 4: Pick the first number, 3. Subtract 3 from the remaining value(s) and divide the outcome by 2. Remove the duplicates and sort:

(19-3)/2 = 8

The resulting list: 3, 8

#### Step 5: Divide all of the remaining even values by 2, remove the duplicates and sort. Repeat the process if there are even numbers in the list:

8/2 = 4

The resulting list: 3, 4

4/2 = 2

The resulting list: 2, 3

2/2 = 1

The resulting list: 1, 3

#### Step 6: Pick the first number, 1. Subtract 1 from the remaining value(s) and divide the outcome by 2. Remove the duplicates and sort:

(3-1)/2 = 1

The resulting list: 1

#### Step 7: Only one number remains, 1. Multiply it by your current GCF:

GCF = 1*1 = 1

The Greatest Common Factor of 19 and 31 is 1