Lowest Common Multiple

Ella has a handful of sweets that she would like to distribute to her friends. She decides to divide these sweets into 3 equal piles. She then changes her mind and decides to divide them into 4 equal piles instead. What is the smallest possible number of sweets Ella could have?

To answer this question, we would need to find the lowest common multiple between 3 and 4. This article will discuss this idea and introduce several methods we can use to find the lowest common multiple for a given set of numbers. All set? Let's begin!

Recap: Multiples and Common Multiples

Before we begin, let us take a quick glance at some earlier topics related to this one, namely multiples and common multiples.

Essentially, \(B\) and \(C\) are multiples of \(A\) if \(B\) and \(C\) are in the multiplication table of \(A\). The multiple of a number, say \(a\), is given by the general formula,

\[\text{Multiple of}\ a=a×z\]

where \(z\in\mathbb{Z}\). Generally speaking, if \(A\times B=C\) then \(A\) and \(B\) are divisors (or factors) of \(C\) or \(C\) is the multiple of both \(A\) and \(B\). We can use the multiplication table to find a particular set of multiples for a given number.

On the other hand:

To find the common multiples for a given set of numbers, you would simply list the multiples of each given number and pick out any identical multiples shared between them. To recall, let us look at an example.

You can find a detailed explanation of multiples and common multiples in the topics, Multiples and Common Multiples, respectively.

The lowest common multiple is a direct result of common multiples. How is that, you may ask? Well, let us return to our previous example.

We found that the common multiples of 2 and 6 are 6 and 12. Comparing these two common multiples, 6 is said to be the lowest common multiple of 2 and 6 since 6 is less than 12. This brings us to the following definition.

The lowest common multiple will be abbreviated as LCM throughout this discussion.

The LCM between two given numbers say \(a\) and \(b\), is denoted by

\[ \text{LCM}(a, b)=c\]

where \(c\) is the LCM. Before we move on to methods for finding the LCM of a given set of numbers, it may be helpful to familiarise yourselves with some of their unique properties.

Properties of LCM

The table below describes two important properties of LCM along with its explanation.

Methods of Finding the LCM

The LCM is always non-zero despite zero being the smallest multiple between any set of numbers. There are three ways in which you can find the LCM for a given set of numbers, namely:

1. Listing method;

2. Prime factorisation method;

3. Division method.

We will now study each method in turn, followed by several worked examples demonstrating their techniques.

Common Multiple Method

This method only requires you to list down the multiples of each given number and identify the lowest common multiple shared between them.

Important note: There is no need for an interval restriction here for a given set of numbers. To find their LCM, simply pinpoint the first smallest common multiple you see that is shared between the numbers is the LCM. You do not have to list everything! Just up to the point for which the multiples are identical.

Here is an example that uses this approach.

Let us now look at one more example.

Although this method seems rather straightforward, it is not as favoured as the other two methods. This method can become rather lengthy and confusing if we were given a set of numbers that have a very large difference between them.

For example, say we are told to find the LCM of 2 and 51. It would take a very long time until we find a common multiple shared between them as we would have to go through all the multiples of 2 before we can hit a number closest to 51.

To resolve this, we would need to use either the prime factorization method or the common divisor method, as we shall soon see in the next two sections.

Prime Factor Method

As the name suggests, the prime factor method is a technique that tells us to break down a given number as the product of its prime factors. You can find a detailed discussion on prime factorisation in the topic: Prime Factorization. There are four steps to this process:

1. Given a set of numbers, write each of them as a product of their prime factors;

2. Express these products in their respective exponent (or index) form;

3. Identify the highest power of all the prime factors for the given numbers;

4. The LCM is the product of the prime factors found in Step 3.

To picture these instructions better, let us observe an example that uses this technique.

Let us now return to our previous conundrum.

Here is one more example.

Common Division Method

There are three steps to this technique:

1. Construct a division table of two columns (as shown below). Place the given numbers in the second column. The first column will be where we put the prime factors.

2. Divide the smallest prime number that can divide at least one of the given numbers and bring down the numbers after this division. The numbers that are not divisible can be brought down as is.

3. Continue this process until the final row has a quotient of 1 for the given numbers.

4. Multiply all the prime factors obtained in the first column. This is the LCM.

The common division method happens to be the most preferred approach of the three methods when finding the LCM. This is because we can simply group all the given numbers in one division table. Here is an example.

We shall look at one more example before we end this section.

LCM vs. HCF

In this segment, we will be making comparisons between the LCM and HCF. Note that the HCF is an abbreviation of the highest common factor. The table below describes each of their notable differences.

You can find a detailed explanation of this in the topic: Highest Common Factor.

Relationship Between LCM and HCF

There are two conditions that tie the concept of the LCM and HCF together, namely

1. The LCM and HCF Formula;

2. The LCM and HCF of Fractions.

In this section, we shall look at each of these ideas in turn with a related example.

The LCM and HCF Formula

This formula states that the product of the LCM and HCF of any two given numbers is always equal to the product of those given numbers. Say we have two numbers, \(a\) and \(b\), then

\[\text{LCM}(a, b)\times \text{HCF}(a, b)=a\times b\]

Note that this formula is only suitable for two given numbers.

The LCM and HCF of Fractions

Say we are given two fractions, \(\frac{a}{b}\) and \(\frac{c}{d}\). We can obtain the LCM and HCF of these two fractions using the following generalized formulas.

\[LCM=\frac{\text{LCM of numerators}}{\text{HCF of denominators}}\]

\[HCF=\frac{\text{HCF of numerators}}{\text{LCM of denominators}}\]

Here is an example.

Examples Involving LCM

We shall conclude this discussion with a few more worked examples that apply the LCM.

Here is another example.

Real-world Examples Using the LCM

The LCM can also be applied to real-world problems, as we have seen at the beginning of our discussion. Let us recall that example and use the idea of an LCM to solve it.

We shall observe one last real-world example to tie up this discussion.