Greatest Common Factor of 54 and 60 is 6

How to find the Greatest Common Factor of 54 and 60?

There are many methods we can apply to calculate the GCF of 54 and 60.


In our first method, we'll find out the prime factorisation of the 54 and 60 numbers.


In our second method, we'll create a list of all the factors of the 54 and 60 numbers.

These are the numbers that divide the 54 and 60 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 54 and 60.

Methods of calculating the GCF of 54 and 60:

Method 1 - Prime Factorisation

With the prime factorisation method, all we have to do is to find the common prime factors of 54 and 60, and then multiply them. Really simple:

Step 1: Let's create a list of all the prime factors of 54 and 60:

Prime factors of 54:

As you can see below, the prime factors of 54 are 2, 3, 3 and 3.

Let's illustrate the prime factorization of 54 in exponential form:

54 = 21x33

Prime factors of 60:

As you can see below, the prime factors of 60 are 2, 2, 3 and 5.

Let's illustrate the prime factorization of 60 in exponential form:

60 = 22x31x51


Step 2: Write down a list of all the common prime factors of 54 and 60:

As seen in the boxes above, the common prime factors of 54 and 60 are 2 and 3.


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:

21x31=6

Done!

According to our calculations above, the Greatest Common Factor of 54 and 60 is 6

Method 2 - List of Factors

With this simple method, we'll need to find all the factors of 54 and 60, 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 54 and 60 without a remainder:

List of factors that divide 54 without a remainder are:

1, 2, 3, 6, 9, 18, 27 and 54.

List of factors that divide 60 without a remainder are:

1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60.


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 54 and 60, we have highlighted the number 6, which means that we have found the Greatest Common Factor, or GCF.

According to our calculations above, the Greatest Common Factor of 54 and 60 is 6

Method 3 - Euclidean algorithm

The Euclidean algorithm says that if number k is the GCM of 54 and 60, then the number k is also the GCM of the division remainder of the numbers 54 and 60.

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:

54, 60

Step 2

Take out, from the set, the smallers number as you divisor: 54

The remaining set is: 60

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

60 mod 54 = 6


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

6, 54

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

Take out, from the set, the smallers number as you divisor: 6

The remaining set is: 54

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

54 mod 6 = 0


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

6

Step 3: Take the remaining number from our set

The Greatest Common Factor of 54 and 60 is 6

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: 54, 60

Step 2: All of the numbers are even.
Divide all of them by 2 and multiply your GCF by 2.
Remove the duplicates and sort.
Repeat the process if all the numbers are even:

The resulting list: 27, 30

GCF = 1*2 = 2


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:

30/2 = 15

The resulting list: 15, 27

Step 4: Pick the first number, 15.
Subtract 15 from the remaining value(s) and divide the outcome by 2.
Remove the duplicates and sort:

(27-15)/2 = 6

The resulting list: 6, 15

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:

6/2 = 3

The resulting list: 3, 15

Step 6: Pick the first number, 3.
Subtract 3 from the remaining value(s) and divide the outcome by 2.
Remove the duplicates and sort:

(15-3)/2 = 6

The resulting list: 3, 6

Step 7: 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

Step 8: Only one number remains, 3.
Multiply it by your current GCF:

GCF = 2*3 = 6

The Greatest Common Factor of 54 and 60 is 6