The GCF (greatest common factor) of the numbers 21 and 24 is 3
On this page we are calculating the Greatest Common Factor of 21 and 24
How to find the Greatest Common Factor of 21 and 24?
There are many methods we can apply to calculate the GCF of 21 and 24.
In our first method, we'll find out the prime factorisation of the 21 and 24 numbers.
In our second method, we'll create a list of all the factors of the 21 and 24 numbers.
These are the numbers that divide the 21 and 24 numbers without a remainder. These are the numbers that divide the 21 and 24 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 21 and 24.
Methods of calculating the GCF of 21 and 24:
Method 1 - Prime Factorisation
With the prime factorisation method, all we have to do is to find the common prime factors of 21 and 24, and then multiply them. Really simple:
With the prime factorisation method, all we have to do is to find the common prime factors of 21 and 24, and then multiply them. Really simple:
Step 1: Let's create a list of all the prime factors of 21 and 24:
Prime factors of 21:
As you can see below, the prime factors of 3 and 7.
Let's illustrate the prime factorization of 21 in exponential form:
Prime factors of 24:
As you can see below, the prime factors of 2, 3, and 3.
Let's illustrate the prime factorization of 24 in exponential form:
Step 2: Write down a list of all the common prime factors of 21 and 24:
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:
Method 2 - List of Factors
With this simple method, we'll need to find all the factors of 21 and 24, 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 21 and 24 without a remainder:
List of factors that divide 21 without a remainder are:
List of factors that divide 24 without a remainder are:
Step 2: Identify the largest common number from the 21 and 24 lists above:
As you can see in the lists of factors from above, for the numbers 21 and 24, we have highlighted the number 3, which means that we have found the Greatest Common Factor, or GCF.
According to our calculations above, the Greatest Common Factor of 21 and 24 is 3
Method 3 - Euclidean algorithm
The Euclidean algorithm says that if number k is the GCM of 21 and 24, then the number k is also the GCM of the division remainder of the numbers 21 and 24.
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:
Step 2
Take out, from the set, the smallers number as you divisor: 21
The remaining set is: 24
Find the reminder of the division between each number and the divisor
Gather the divisor and all of the remainders and sort them in ascending order. Remove any duplicates and 0. Our set is: 3,21
Repeat the process until there is only one number in the set.
Take out, from the set, the smallers number as you divisor: 3
The remaining set is: 21
Find the reminder of the division between each number and the divisor
Gather the divisor and all of the remainders and sort them in ascending order. Remove any duplicates and 0. Our set is: 3
Repeat the process until there is only one number in the set.
Step 3: Take the remaining number from our set
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
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:
The resulting list: 12, 21
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:
The resulting list: 6, 21
Step 4: 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:
The resulting list: 3, 21
Step 5: Pick the first number, 3. Subtract 3 from the remaining value(s) and divide the outcome by 2. Remove the duplicates and sort:
The resulting list: 3, 9
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:
The resulting list: 3