What is the greatest common factor of 26 and 39

How to find the Greatest Common Factor of 26 and 39?

There are many methods we can apply to calculate the GCF of 26 and 39.

In our first method, we'll find out the prime factorisation of the 26 and 39 numbers.

In our second method, we'll create a list of all the factors of the 26 and 39 numbers.

These are the numbers that divide the 26 and 39 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 26 and 39.

• Prime Factorisation
• List of factors
• Euclidean algorithm
• Binary Greatest Common Divisor algorithm

Method 1 - Prime Factorisation

With the prime factorisation method, all we have to do is to find the common prime factors of 26 and 39, and then multiply them. Really simple:

Step 1: Let's create a list of all the prime factors of 26 and 39:

Prime factors of 26:

As you can see below, the prime factors of 26 are 2 and 13.

Let's illustrate the prime factorization of 26 in exponential form:

26 = 21x131

Prime factors of 39:

As you can see below, the prime factors of 39 are 3 and 13.

Let's illustrate the prime factorization of 39 in exponential form:

39 = 31x131

Step 2: Write down a list of all the common prime factors of 26 and 39:

As seen in the boxes above, the common prime factors of 26 and 39 are 13.

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:

131=13

Done!

According to our calculations above, the Greatest Common Factor of 26 and 39 is 13

Method 2 - List of Factors

With this simple method, we'll need to find all the factors of 26 and 39, 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 26 and 39 without a remainder:

List of factors that divide 26 without a remainder are:

1, 2, 13 and 26.

List of factors that divide 39 without a remainder are:

1, 3, 13 and 39.

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 26 and 39, we have highlighted the number 13, which means that we have found the Greatest Common Factor, or GCF.

According to our calculations above, the Greatest Common Factor of 26 and 39 is 13

Method 3 - Euclidean algorithm

The Euclidean algorithm says that if number k is the GCM of 26 and 39, then the number k is also the GCM of the division remainder of the numbers 26 and 39.

We follow this procedure until the reminder is 0.

The Greatest Common Divisor is the last nonzero number.

Step 1: Sort the numbers into ascending order:

26, 39

Step 2

Take out, from the set, the smallers number as you divisor: 26

The remaining set is: 39

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

39 mod 26 = 13

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

13, 26

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

Take out, from the set, the smallers number as you divisor: 13

The remaining set is: 26

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

26 mod 13 = 0

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

13

Step 3: Take the remaining number from our set

The Greatest Common Factor of 26 and 39 is 13

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.

Step 1: Sort the numbers, and set initial GCF equal to 1

The list: 26, 39

Step 2: 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:

26/2 = 13

The resulting list: 13, 39

Step 3: Pick the first number, 13.
Subtract 13 from the remaining value(s) and divide the outcome by 2.
Remove the duplicates and sort:

(39-13)/2 = 13

The resulting list: 13

Step 4: Only one number remains, 13.
Multiply it by your current GCF:

GCF = 1*13 = 13

The Greatest Common Factor of 26 and 39 is 13

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