Prime Factorization

Prime Factorization means to find the factors of a number such that the factors are divisible by 1 and itself.


Given number when divided into numbers that doesn’t contain remainder are called factors.

Example: Say you want to find the factors of 24. From the test of divisibility of 2, if the digit at the unit place is 0,2,4,6,8 then the number is divisible by 2. Here the number is 24 that has a unit place digit is 4. Therefore the number is divisible by 2.

24 = 2 x 12

which means 2 and 12 are factors of 24.

From the divisibility test of 3 addition of digits if divisible by 3 then the given number is divisible by 3.

24 has digits 2 and 4. Addition of these digits is 2 + 4 = 6 which is divisible by 3

therefore 24 is divisible by 3

24 = 3 * 8

3 and 8 are factors of 24.

we can divide the number further and find more factors

Prime Numbers

Prime numbers are the numbers divisible by 1 and itself. Prime numbers are integers that are greater than 1, that have as factors only 1 and the number itself. The prime numbers under 100 are as follows:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

If you consider any number from above say 41, it is divisible by 1 and 41 only. There are no other numbers that divide into 41 evenly.

The number 2 is considered the first prime number. Again, it is divisible only by 1 and itself. Interestingly, 2 is the only even number that is prime.

This is because the other even numbers greater than 2 are 4,6,8,10,12,14,16,18…. all these numbers end with 0,2,4,6,8. As per divisibility of 2, they are divisible by 2 and so cannot be called prime. They will minimally have factors 1, 2, and the number itself and therefore they fail to meet the definition of a prime number.

Some odd numbers are prime but not all. Odd numbers are 1,3,5,7,9,11,13,15…. In these numbers 9 and 15 are divisible by 3 and 15 is also divisible by 5 and so fails to be prime number.

Composite Numbers

Numbers that are not prime, are called composite (com•POS’it) numbers.

A good example of a composite number is 24. Apart from 1 and 24 it is divisible by 2,3,4,6,8,12 as well.

Another good example of a composite number is 50. You can certainly divide 50 by 1 and 50 to get to 50. But you can ALSO divide by 2, 5, 10, and 25

Prime Factorization:

When a number is written as the product of all its prime factors, we call that “prime factorization.” To find the prime factors we can use a “factor tree”.

Let’s factorize 120. We can choose any two factors we desire so let’s start with 30 • 4 = 120. Currently, our factor tree looks like this.

prime factorization
prime factorization

Next, we can factor 30 as 6•5 and 4 as 2•2, so our factor tree now looks like this:

factors of 120
factors of 120

Finally, we can continue to factor the 6 as 2•3, but the factors at the other ends of the tree, 5, 2, and 2, are all prime and cannot be further factored. Thus, the final factor tree is:

factorizing 120
factorizing 120


Prime Number: A prime number is a whole number that only has two factors which are itself and one. 2 is the first prime number.

Composite Number: A composite number has factors in addition to one and itself. In other words, a composite number is not a prime number.

0 and 1: Considered neither prime nor composite.

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