Factors, Prime & Composite Numbers
In this course, we will cover the fundamentals of index notation, including how to find the nth root of a number and how to combine multiple indices without using the laws of indices.
Students will also learn the definition of factors, prime and composite numbers, and how to break down any composite number into its prime factors using both the long division method and the preferred tree method.
This foundation course prepares students to confidently understand and apply the concepts in the topic S1 Factors & Multiples to find the HCF and LCM of a group of numbers in the main EMath GCE O-Level course.
What Is Index Notation/Form?
First, let us explore the concept of index notation, which is a concise way of representing repeated multiplication of the same number.
Index notation/form is a short way of writing a number being multiplied by itself several times. In this example, “2” multiply itself 5 times, and we can write in index form:
\(2\times2\times2\times2\times2=2^5\)
Where “2” is the base, and “5” is the power.
How To Evaluate The Nth Root?
To find the nth root of a number in index form, we simply divide the power by the value m indicated on the root symbol. For example:
\(\sqrt[m]{5^n} = 5^{\frac{n}{m}}\)
For example, to perform a square root of prime factors, we divide the power by 2:
\(\sqrt{2^4\times3^6}=2^\frac{4}{2}\times3^\frac{6}{2}=2^2\times3^3 \)
To perform a cube root of prime factors, we divide the power by 3:
\(\sqrt[3]{2^9\times3^6}=2^\frac{9}{3}\times3^\frac{6}{3}=2^3\times3^2\)
How To Combine Indices Together
For students who have not yet learned the Law of Indices, this lesson explains how to combine expressions with the same base without formally applying the law. Students who have already learned the Law of Indices may skip this video.
When two indices with the same base are multiplied together, their powers are added. In this example, both terms are base “2”, so we can add the power:
\[
\begin{align*}
2^3\times2^5&=2^{3+5}\\
&=2^8\\
\end{align*}
\]
What Is Factoring & What Are Factors?
The process of factoring is taking a number apart to find the factors. Factors are numbers that divide another number equally with no reminder.
To find the factors of 24:
- 1×24
- 2×12
- 3×8
- 4×6
Therefore, the factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24.
To find the factors of 30:
- 1×30
- 2×15
- 3×10
- 5×6
Therefore, the factors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30.
What Are Prime & Composite Numbers?
Next, we will learn the definitions of prime numbers and composite numbers, and understand what distinguishes one from the other.
- A prime number is a number that has exactly two factors: 1 and itself.
- Key prime numbers to remember includes 2, 3, 5, 7, 11, 13, 17, 19, and 23.
- Composite numbers are numbers that have more than two factors.
- The number “1” is neither prime nor composite, as it has only one factor.
Two Prime Factorisation Methods
Now, to break down any composite number into its prime factors , we can use two different methods:
- the long division method
- the tree method (preferred)
The tree method is generally preferred as it is less tedious and allows large composite numbers to be broken down more quickly as compared to the long division method.
For example, to break down 1620 as prime factors, since the number ends with a “0”, we can break down 1620 into 162 multiply by 10. With a smaller number, we can obtain the final prime factors very quickly.
Both methods will be cover in detailed in the video at the end of the chapter.
Watch Full Concept Breakdown
In this video, we bring together all the key ideas covered above, including additional explanation & examples. Students are encouraged to view the entire video, as it reinforces student’s understanding of the concepts.