Factors & Multiples
In this lesson, students will learn the concepts of how to find the Highest Common Factor (HCF) and Lowest Common Multiple (LCM).
Two methods for finding the HCF and LCM of a group of numbers are covered: the long division method and the selection method (preferred), followed by worked examples using the selection method.
This lesson builds on the Factors, Prime and Composite Numbers topic in the EMaths Foundation Course, and students are encouraged to revisit it if they need a refresher on definitions, prime factorisation, indices, and related number manipulation skills.
Related Lessons:
What Is HCF?
First, let us understand the concept of the Highest Common Factor (HCF) by finding the HCF of 28 and 42. Using the factoring method, we break each number down into its factors and identify the largest factor common to both numbers.
- Factors of 28 are 1, 2, 4, 7, 14 and 28
- Factors of 42 are 1, 2, 3, 6, 7, 14, 21 and 42
Since 14 is the smallest common factor for 28 and 42, the HCF of 28 and 42 is 14.
What Is LCM?
Next, let us learn about the concept of the Lowest Common Multiple (LCM) by finding the LCM of 12 and 20. Using the multiple method, the multiples are:
- Multiples of 12 are 12, 24, 36, 48, 60, 72 ….
- Multiples of 20 are 20, 40, 60, 80, 100, 120 ….
Since 60 is the smallest number that is a multiple of both 12 and 20, the LCM of 12 and 20 is 60.
Finding The HCF & LCM Using Long Division Method
Now, the long division method is the most commonly taught method in most schools. We will be covering this method in the video as the end of this chapter as it is not the preferred method.
When using the long division method, note the following important points:
- The HCF is always found first during the division process.
- The LCM is found at the end, after all common factors have been divided out.
Why is the long division method not recommended?
Firstly, when students are given a pair of large numbers — for example 12,600 and 18,000 — find the HCF & LCM using the long division method can become extremely lengthy and time-consuming.
Secondly, in examination questions where numbers are already expressed in index notation, it is difficult to apply division directly to numbers written in powers.
Lastly, at times we are required to find only the LCM, but the long division method always start off by finding the HCF, a redundant working.
Finding The HCF & LCM Using Selection Method
The preferred method is to find the HCF and LCM using the selection method. It is the preferred approach, especially for handling large numbers that are difficult to manage using long division.
In summary, the method is:
- To find the HCF, select only the common factors and choose the smallest power of each common prime factor.
- To find the LCM, select all required prime factors and choose the largest power of each prime factor.
For example, the prime factors of of 126 and 180 are:
\[
\begin{align*}
126&=2\times3^2\times7\\
180&=2^2\times3^2\times5\\
\end{align*}
\]
Therefore:
- The HCF is 18 since the smallest common prime factors are 2 and \(3^2\)
- The LCM is 1260 since the largest for all prime factors are \(2^2\), \(3^3\), 5 and 7
Watch Full Concept Breakdown With Examples
In this video, we will cover all the concepts taught above, including 3 carefully selected examples to find the HCF and LCM using the selection method at the end of the video. The final example is especially interesting as it involves 3 pairs of composite numbers.
You may wish to try these questions on your own before watching the video solutions.
- Find the HCF & LCM of \(2^2\times3^3\times5^5\times7^2\) and \(2\times5^3\times7^4\)
- Find the HCF & LCM of \(2\times3^3\times5^5\times7^2\) and \(2\times3^2\)
- Find the HCF & LCM of \(2\times3\times5^2\times7^{12}\), \(3^3\times5^3\times7^{14}\) and \(2\times3\times5\times7^2\)