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Prime Numbers and Prime Factorization

Updated: Jun 22, 2021


Dear Secondary Math students, today, we will be covering the topic on Prime Factorization. We will be teaching students concepts and how to do questions regarding prime factorization, Highest Common Factor (HCF) and Lowest Common Multiple (LCM). Let's begin!



In this note, you will learn:


1. What are prime numbers?

2. What is prime factorization? 3. What is Highest Common Factor (HCF) and Lowest Common Multiple (LCM)?

1. What are prime numbers?


The definition of a “prime number” is given to be a natural (one used for counting and ordering) number that is greater than 1, and not a result of a product of two smaller natural numbers.



The general rule of thumb to know if a number is prime is that 1) it must be greater than 1, and 2) can only be divided by 1 or itself. Examples are 2,3,5,7,9,11,13… so on and so forth.



2. What is prime factorization?


Now that you know what prime numbers are, next we will be learning about prime factorization. What exactly is prime factorization?




Prime factorization is the decomposition of a composite number into a product of smaller prime numbers. In other words, it can also be said to be the multiplication of prime numbers to produce the original number. So, how do we do prime factorization?


Let’s take a look at an example below:


Let’s say we are given a number: 840, and we are told to dissect it to be the form of a prime factorization. All we have to do is to split this up into the multiplication of prime numbers until it reaches the smallest prime number possible!



To begin, we will start from the smallest prime number and work our way up, which means that since 840 is an even number, it can be divisible by 2, which is the only even prime number.


840 / 2 = 420


We will have to repeat these two more times, and we will end up with 105. Now, 105 is an odd number and can no longer be divisible by 2, hence we must increase to the next prime number that is divisible by 105, which is 3. That will give us 35, indivisible by 3 and so we will increase to the next prime number to 5, which we ended up with 7. Since 7 is only divisible by 1 and itself, it is hence a prime number we have reached after all the division.



Therefore,


840 = 2 x 2 x 2 x 3 x 5 x 7

= 23 x 3 x 5 x 7


And that is how we do prime factorization!



Question time!


Given the numbers below, prime factorize it and leave your answers in the comments down below!


1. 3465

2. 241920


3. 1488375


3. What is Highest Common Factor (HCF) and Lowest Common Multiple (LCM)?


Now that you know how prime factorization works, let’s learn about the Highest Common Factor, also known as HCF and the Lowest Common Multiple, also know as LCM.




The Highest Common Factor (HCF) of two or more integers, which cannot all be zero, is the greatest positive integer the group of integers can be divisible by. So, how do we find the HCF of two or more integers? Let’s take a look at the example below:



Given that we have the numbers: 12, 30 and 84. First of all, let’s prime factorize them.

12 = 2 x 2 x 3 = 2^2 x 3

30 = 2 x 3 x 5 = 2 x 3 x 5

84 = 2 x 2 x 3 x 7 = 2^2 x 3 x 7



Now, what we must do is to extract out only the least quantity of common prime factors and for prime factors that are unique to their own integers respectively is not taken. that exist in them. If you take a closer look, you will realize that common prime numbers that exist between them are one ‘2’ and one ‘3’.


12 = 2 x 2 x 3

30 = 2 x 3 x 5

84 = 2 x 2 x 3 x 7


Therefore, our HCF for these set of numbers: 12, 30 and 84 is 2 x 3 = 6

Next, we will be learning about the Lowest Common Multiple between two integers.



Lowest Common Multiple of two integers, or LCM, is the smallest positive integer that can be divisible by both the integers. Let’s take a look at the example shown below:



Given that we have the numbers: 108 and 280. The method to find the smallest positive integer that is divisible by both the integers is to extract out the greatest quantity of common prime factors and the prime factors that are unique to a particular number only. Let’s see how it’s done:



First, let’s prime factorize the two integers:

108 = 2 x 2 x 3 x 3 x 3 = 2^2 x 3^3

840 = 2 x 2 x 2 x 3 x 5 x 7 = 2^3 x 3 x 5 x 7



What we want is the greatest quantity of common prime factors and prime factors that are unique to a particular number only, which means that we will extract out three ‘2’s instead of two ‘2’s from the integer 840, three ‘3’s instead of one ‘3’ from integer 108, and both one ‘5’ and one ‘7’ from integer 840. This will give us:


LCM = 23 x 33 x 5 x 7 = 7560


However, there is another method to do this, which is more systematic and easier to comprehend. It is called the Common Division Method.



As you can see from the diagram above,


Firstly, you draw a table to include both the integers which LCM you are trying to find at the top. Then, start dividing it by the smallest prime factor to both the numbers until BOTH numbers are no longer divisible by that prime factor then increase the value of the prime factor.



The column on the most left-hand side will be the prime factors that makes up your LCM of the two integers. Both the numbers should be divided until they give a value of 1. Lastly, multiply all the prime factors you have gotten from the left column and you will get the LCM of the two numbers.


And that’s how you solve for both HCF and LCM!



Question Time!


1. Written as the product of its prime factors 5430 = 2 x 3 x 5 x 181,

a) Express 4320 in its product of prime factors

b) Hence write down the LCM of 5430 and 4320, giving your answer in the form of product of prime factors.


2. When written as a product of its prime factors,

p = 2 x 3 x 5^2

q = 2^3 x 3^6 x 5^9

r = 2^5 x 3^3 x 5


a) Find the value of the cube root of q

b) Find the LCM of p, q and r, giving your answers in the form of its prime factors

c) The greatest number that will divide p, q and r exactly


3) Written as a product of its prime factors, 2250= 2 x 3^x x 5^y

a) Find the values of x and y

b) Hence, find the smallest positive integer k such that 2250/k is a square number

And that’s all we have for today, students! Math Lobby hopes that you have a new and clear understanding of prime numbers, prime factorization, HCF and LCM. If you have any questions or inquiries, feel free to contact Math Lobby on our website, Facebook or Instagram page!


As always: Work hard, stay motivated and I wish all students a successful and enjoyable journey with Math Lobby!



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