Let us have 2 fruit baskets, the first basket contains orange, apple, banana, papaya and the second basket contains kiwi, apple, guava, banana. So the common fruits in both baskets are apples and bananas. Similarly in set A we have even numbers from 1 to 10 and in set B we have numbers divisible by 4. Now can you pick the common numbers in them? Yes, it’s 4 and 8.

Now the biggest common number is 8. This biggest common number is called the Greatest Common Factor (gcf). It is also known as Greatest common divisor or highest common factor.

Prime Factorization

Take 4 apples, you can share these apples with 2 or 4 kids equally or you can also eat them all.  So the factors of 4 are 1, 2 and 4. But if you have to represent these factors in prime numbers then 4 = 2 ✕ 2. So the prime factor of 4 is 2². Hence Prime factorization is a method of writing  numbers(composite numbers) as a product of prime numbers.

Ex: 36 = 9 ✕ 4 = 3² ✕ 2². So the prime factors of 36 = 3² ✕ 2²

Common Methods of Prime Factorization

Factor tree method and division methods are 2 most commonly used prime factorization techniques. 

1. Factor tree technique – The given number is factorized until you get the factors as prime factors. The given number will be the root and the factors are represented as branches of the tree.

Ex. 

So the prime factors of 100 are 2² 5²

2. Division technique – Here the given number is divided using prime numbers until you get the quotient as 1.

Ex. 100 is divided by 2, so 1002 = 50 Now, 502 = 25,

but 25 cannot be divided by 2 so we divide it by 5. ∴ 25 5 = 5 and 5 5 = 1

Hence the prime factors of 100 are 2² and 5².

Can We Use a Prime Factorization Method for Finding GCF?

Yes. Prime factorization method is also one of the methods used to find GCF.

Ex. Take 15 and 20. Prime factors of 15 are 3 and 5 that of 20 are 5 and 2². Hence the GCF of these 2 numbers is 5.   

There are 2 more methods for finding GCF, they are division method and listing common factors method. Division method is the same as that we used in prime factorization. After dividing the given numbers we find the factors and then we take out the biggest common factor.

In listing common factor methods we list out the common factors of given numbers after factorization. Then we take out the GCF. For smaller numbers we use division techniques but for bigger numbers we use factorization techniques.

Applications of GCF and Prime Factorization

Gcf is mainly useful for finding fractions in maths, in the construction field for measurements, while equally distributing the things, etc. Prime factorization is quite handy in cryptography. It is a study of secret codes. It is largely beneficial in the mathematical field as it is the fundamental theorem of arithmetics.

Now you find GCF and prime factors for the below group of numbers.

a) 12, 16 b) 32, 40, 54, 18

a)  Prime factors of 12 = 4 3 = 2² 3

     Prime factors of 16 = 4 4 = 2⁴

     ∴ GCF of 12, 16 is 4 

b)  Prime factors of 32 = 16 2 = 4² 2 = 2⁵

     Prime factors of 40 = 8 5 = 2³ 5

     Prime factors of 54 = 9 6 = 3² 3 2 = 3³ 2

     Prime factors of 18 = 9 2 = 3² 2 

     ∴ GCF of 32, 40, 54, 18 is 2.

Recommendation: To Ease up the GCF calculation of the above values, you can also use the greatest common factor calculator.