How to find the Greatest Common Factor of 30 and 42?
There are many methods we can apply to calculate the GCF of 30 and 42.
In our first method, we'll find out the prime factorisation of the 30 and 42 numbers.
In our second method, we'll create a list of all the factors of the 30 and 42 numbers.
These are the numbers that divide the 30 and 42 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 30 and 42.
Methods of calculating the GCF of 30 and 42:
Method 1 - Prime Factorisation
With the prime factorisation method, all we have to do is to find the common prime factors of 30 and 42, and then multiply them. Really simple:
Step 1: Let's create a list of all the prime factors of 30 and 42:
Prime factors of 30:
As you can see below, the prime factors of 30 are 2, 3 and 5.
Let's illustrate the prime factorization of 30 in exponential form:
30 = 21x31x51
Prime factors of 42:
As you can see below, the prime factors of 42 are 2, 3 and 7.
Let's illustrate the prime factorization of 42 in exponential form:
42 = 21x31x71
Step 2: Write down a list of all the common prime factors of 30 and 42:
As seen in the boxes above, the common prime factors of 30 and 42 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 30 and 42 is 6
Method 2 - List of Factors
With this simple method, we'll need to find all the factors of 30 and 42, 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 30 and 42 without a remainder:
List of factors that divide 30 without a remainder are:
1, 2, 3, 5, 6, 10, 15 and 30.
List of factors that divide 42 without a remainder are:
1, 2, 3, 6, 7, 14, 21 and 42.
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 30 and 42, 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 30 and 42 is 6
Method 3 - Euclidean algorithm
The Euclidean algorithm says that if number k is the GCM of 30 and 42, then the number k is also the GCM of the division remainder of the numbers 30 and 42.
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:
30, 42
Step 2
Take out, from the set, the smallers number as you divisor: 30
The remaining set is: 42
Find the reminder of the division between the number and the divisor
42 mod 30 = 12
Gather the divisor and all of the remainders and sort them in ascending order. Remove any duplicates and 0. Our set is:
12, 30
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: 30
Find the reminder of the division between the number and the divisor
30 mod 12 = 6
Gather the divisor and all of the remainders and sort them in ascending order. Remove any duplicates and 0. Our set is:
6, 12
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: 12
Find the reminder of the division between the number and the divisor
12 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 30 and 42 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: 30, 42
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: 15, 21
GCF = 1*2 = 2
Step 3: Pick the first number, 15.
Subtract 15 from the remaining value(s) and divide the outcome by 2.
Remove the duplicates and sort:
(21-15)/2 = 3
The resulting list: 3, 15
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:
(15-3)/2 = 6
The resulting list: 3, 6
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
Step 6: Only one number remains, 3.
Multiply it by your current GCF:
GCF = 2*3 = 6
The Greatest Common Factor of 30 and 42 is 6