Preparing for your GCE O Level E-math exams requires a thorough understanding of essential numerical concepts that form the foundation of your mathematical knowledge. Among these crucial topics are Prime Numbers, Highest Common Factor (HCF), Lowest Common Multiple (LCM), Perfect Squares, and Perfect Cubes. These topics are the cornerstones for more advanced algebraic concepts that you’ll face in the future.

In this article, we will provide you with a summarised conceptual guide on Prime Numbers, HCF, and LCM, as well as some application questions that you can try your hand at.

## Categorising Natural Numbers using Prime Factorisation

Natural numbers (also known as positive integers) such as 1, 2, 3, 4 can be categorised into three main groups: the number one (1) itself, composite numbers and prime numbers.

### Composite Numbers

Composite numbers have more than two factors and can be divided by numerical values other than 1 and themselves. Examples include 4, 6, 8, 9, 10, 12, 14 and 15.

### Prime Numbers

Prime numbers, on the other hand, have exactly two factors (1 and itself) and cannot be divided evenly. Examples include 2, 3, 5, 7, 11, 13 and 17.

Fun Fact: The smallest prime number is 2!

**Checkpoint!**** Prime ****Numbers****: Question 1**

Can you identify the prime numbers between 1 to 30?

**Check your answers!**** Did you identify all the prime ****numbers ****correctly?**

### Prime Numbers: Application Questions

**Checkpoint!**** Prime ****Numbers****: Question 2**

Express 4840 in index notation.

**Check your answers!** Did you get it right?

## Highest Common Factor (HCF): Finding the Greatest Common Factor

HCF is defined as the greatest common factor between two or more numbers; the largest positive integer that divides the numbers without leaving a remainder.

Take 8 and 12, for instance.

The HCF of 8 and 12 will be 4 as 4 is the highest value that can divide both 8 and 12.

Now, let’s understand how we can find the HCF of the following values in index notation using this example below.

**Find the highest common factor of the following ****values ****in index notation.**

**924 = 2^2 x 3 x 7 x 11**

**2520 = 2^3 x 3^2 x 5 x 7**

**2548 = 2^2 x 7^2 x 13**

**Checkpoint!**** HCF: Question 1**

Find the highest common factor (HCF) of 820, 2120, and 2240 using prime factorisation.

**Check your answers!**** Did you get it right?**

## Lowest Common Multiple (LCM): Calculating the Lowest Common Multiple

The lowest common multiple of two numbers, for example: x and y, is denoted by LCM(x,y). The LCM is the smallest integer that is divisible by both x and y.

Let’s put this into numerical terms for better understanding. Let’s take numbers 4 and 6, for example.

The multiples of the number 4 are 4, 8, 12, 16, 20, 24, and so on.

The multiples of the number 6 are 6, 12, 18, 24, and so on.

Of this, the common multiples for 4 and 6 would be 12, 24, 36, and so on. The least common multiple amongst these numbers is 12.

Now that you understand what LCM is about, let’s try to find the LCM of 45 and 60 using the prime factorisation method.

Now, it’s your turn!

**Checkpoint!**** LCM: Question 1**

Find the lowest common multiple of the following values.

**924 = 2^2x 3 x 7 x 11**

**2520 = 2^3 x 3^2 x 5 x 7**

**2548 = 2^2 x 7^2 x 13**

**Check your answers!**** Did you get it right?**

## Perfect Squares

A perfect square is defined as an integer that can be expressed as the square of another integer. In other words, a perfect square is the result of multiplying an integer by itself.

For example, take a look at the numbers 4, 9, 16 and 25.

This set of numbers can be classified as perfect squares as they can be written as:

**4 = 2^2 = 2 x 2**

**9 = 3^2 = 3 x 3**

**16 = 4^2 = 4 x 4**

**25 = 5^2 = 5 x 5**

Therefore, we can conclude that if *n* is an integer, then n^2 is a perfect square.

Now, let’s solve an example question together!

**Find the smallest value of ***k*** such that 2520***k*** is a perfect square.**

Now, try a similar question below.

**Checkpoint!**** Perfect Square: Question 1**

Find the smallest value of *k* such that square root 2520k is a whole number.

**Check your answers!**** Did you get it right?**

## Perfect Cubes in Numbers

A perfect cube is defined as a number that can be expressed as the cube of an integer. Specifically, if x is a perfect cube of y, then x=y^3.

Take the numbers: 8, 27, 64 and 125 for example. They are perfect cubes because they can be represented as:

**8 = 2^3 = 2 x 2 x 2**

**27 = 3^2 = 3 x 3 x 3**

**64 = 4^3 = 4 x 4 x 4**

**125 = 5^3 = 5 x 5 x 5**

Moreover, when we take the cube root of a perfect cube, we will obtain a natural number and not a fraction; cube root x = y.

Have you understood the concept of Perfect Cubes so far?

If so, let’s jump right into an example question.

**Find the smallest value of ***k*** such that ****2520k**** is a perfect cube.**

Now, try a similar question below.

**Checkpoint! Perfect Square: Question 1**** **

Question 1: Find the smallest value of *k* such that cube root 2520k is a whole number.

**Check your answers!**** Did you get it right?**

## Application Questions

Now that you’ve understood the concepts of prime numbers, HCF, LCM, perfect squares and perfect cubes, try out this application question.

**Checkpoint!**** Application Question 1**

Find the smallest positive integer *k* such that 392*k *is a multiple of 396.

**Check your answers!**** Did you get it right?**

## Still want more?

Have you understood the concepts of Prime Numbers, HCF and LCM? Now, test your knowledge by attempting this word problem!

Still want more? Then, join our interactive live teaching sessions, where we explore various topics covered in the GCE O Level Elementary Mathematics exams, including interesting subjects like Prime Numbers, HCF and LCM and more.

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