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?

Lowest Common Multiple Lowest Common Multiple

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    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.

    A multiple of a non-zero integer \(A\) is a non-zero integer \(C\) that can be obtained by multiplying it with another number, say \(B\).

    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:

    A common multiple is a multiple shared between two (or more) numbers.

    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.

    Find the first 8 multiples of 2 and 6. Do they share any common multiples for this given range?

    Solution

    The table below shows the first 8 (non-zero) multiples of 2 and 6.

    NumberFirst 8 multiples
    22, 4, 6, 8, 10, 12, 14, 16
    66, 12, 18, 24, 30, 36, 42, 48

    The table above shows that the common multiples of 2 and 6 for this interval are 6 and 12.

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

    What is the Lowest Common Multiple?

    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 (LCM) is the smallest common multiple shared between two (or more) numbers.

    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.

    Property

    Description and Example

    The LCM of two prime numbers is always the product of the given numbers.

    This pair of prime numbers are often referred to as co-primes. Since every prime number has two factors: 1 and the number itself, the only common factor of any co-prime is 1.The general formula for this concept states that if \(a\) and \(b\) are co-primes, then the LCM of \(a\) and \(b\) is equal to \(\text{LCM}(a,b)=a\times b\).

    For example, given two co-primes 2 and 7, the LCM of 2 and 7 is 14 since

    \[\text{LCM}(2,7)=2\times 7=14\]

    The LCM of any set of numbers is never less than the given numbers.

    Given the definition, the LCM is the smallest number that a given set of numbers divide into. Oftentimes, the LCM will be greater than both or at least one of the given numbers. For example, the LCM of 3 and 4 is 12, which is indeed much greater than the given numbers. For a special case, given a set of numbers where one of them is a factor of the other, the LCM is greater than the number itself. For example, say we are told to find the LCM of 9 and 18. Since 9 is a factor of 18, then 18 is the LCM of 9 and 18.

    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.

    Find the LCM of 18 and 27.

    Solution

    Let us begin by listing the first few (non-zero) multiples of 18 and 27.

    Given a set of numbers, list their multiples to the point where you find the first common multiple shared between them. There is no need for you to list the multiples beyond that as you only need to find the smallest possible multiple shared between them.

    Multiples of 18: 18, 36, 54, 72,...

    Multiples of 27: 27, 54, 81,...

    From the lists above, we find that the LCM of 18 and 27 is 54. Thus, \(\text{LCM}(18, 27)=54\).

    Let us now look at one more example.

    Identify the LCM of 15, 20 and 25.

    Solution

    As before, we shall list down the first few multiples of 15, 20 and 25.

    Multiples of 15: 15, 30, 45, 60, 75, 90, 105, 120, 135, 150, 165, 180, 195, 210, 225, 240, 255, 270, 285, 300,...

    Multiples of 20: 20, 40, 60, 80, 100, 120, 140, 160, 180, 200, 220, 240, 260, 280, 300, 320,...

    Multiples of 25: 25, 50, 75, 100, 125, 150, 175, 200, 225, 250, 275, 300, 325,...

    Here, we see that the LCM of 15, 20 and 25 is 300 and so \(\text{LCM}(15, 20, 25)=300\).

    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.

    Find the LCM of 16, 24 and 28.

    Solution

    To begin, let us first break down each of these given numbers as a product of its prime factors.

    \(16=2\times 2\times 2\times 2\)

    \(24=2\times 2\times 2\times 3\)

    \(28=2\times 2\times 7\)

    We will now write these products in their exponent form.

    \(16=2^4\)

    \(24=2^3\times 3\)

    \(28=2^2\times 7\)

    From here, notice that the prime factors are 2, 3 and 7 with 4, 1 and 1 being their highest power respectively. Thus, the LCM is

    \(\text{LCM}(16, 24, 28)=2^4\times 3\times 7=336\)

    Let us now return to our previous conundrum.

    What is the LCM of 2 and 51?

    Solution

    Following our previous example, we need to break down 2 and 51 as a product of their prime factors.

    \(2=2\)

    \(51=3\times 17\)

    Expressing these products in their exponent form yields

    \(2=2^1\)

    \(51=3^1\times 17^1\)

    The prime factors, in this case, are 2, 3 and 17. The highest power for each of these prime factors is all 1. Hence, the LCM is

    \(\text{LCM}(2, 51)=2\times 3\times 17=102\)

    Here is one more example.

    Find the LCM of 63, 125 and 245.

    Solution

    As before, let us start by breaking down each of these given numbers as a product of its prime factors.

    \(63=3\times 3\times 7\)

    \(125=5\times 5\times 5\)

    \(245=5\times 7\times 7\)

    We will now express these products in their exponent form.

    \(63=3^2\times 7\)

    \(125=5^3\)

    \(245=5\times 7^2\)

    The prime factors, in this case, are 3, 5 and 7 with 2, 3 and 2 being their highest power respectively. Thus, the LCM is

    \(\text{LCM}(63, 125, 245)=3^2\times 5^3\times 7^2=55125\)

    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.

    Prime Factors

    Given Numbers

    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.

    What is the LCM of 60, 72 and 84?

    Solution

    Let us now construct the division table.

    60, 72, 84

    We shall now start our division process. Notice that 2 is the smallest prime that is divisible by all three given numbers. Remember to bring down the resulting quotient!

    260, 72, 84
    30, 36, 42

    Again, we can divide these numbers by 2.

    260, 72, 84
    230, 36, 42
    15, 18, 21

    Now notice that 18 is the only number divisible by 2. So we will only have a resulting quotient for the number 18. Numbers 15 and 21 will be brought down as is.

    260, 72, 84
    230, 36, 42
    215, 18, 21
    15, 9, 21

    Let us now continue this process until our last row has a quotient of 1 for all the given numbers.

    260, 72, 84
    230, 36, 42
    215, 18, 21
    315, 9, 21
    35, 3, 7
    55, 1, 7
    71, 1, 7
    1, 1, 1

    Voila! We have a complete division table. We shall now calculate the LCM by multiplying all the prime factors found on the left-hand side of the table above.

    \(\text{LCM}(60, 72, 84)=2\times 2\times 2\times 3\times 3\times 5\times 7=2^3\times 3^2\times 5\times 7=2520\)

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

    What is the LCM of 18, 39 and 42?

    Solution

    As before, we shall construct a division table and divide these given numbers by the smallest prime number until the final row has a quotient of all 1.

    218, 39, 42
    39, 39, 21
    33, 13, 7
    71, 13, 7
    131, 13, 1
    1, 1, 1

    To calculate the LCM, we will multiply all the prime factors obtained in the first column of the table above.

    \(\text{LCM}(18, 39, 42)=2\times 3\times 3\times 7\times 13=2\times 3^2\times 7\times 13=1638\)

    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.

    Lowest Common Multiple (LCM)

    Highest Common Factor (HCF)

    The LCM is the smallest common multiple shared between a given set of numbers.

    The HCF is the largest common factor shared between a given set of numbers.

    The LCM for a given set of numbers is always greater than (or equal to) the given numbers.

    The HCF for a given set of numbers is always lesser than (or equal to) the given numbers.

    The LCM is the smallest number that a given set of numbers divide into.

    The HCF is the largest number that divides a given set of numbers without leaving a remainder.

    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.

    Given the numbers 6 and 8, the LCM and HCF are

    \[\text{LCM}(6, 8)=24\]

    \[\text{HCF}(6, 8)=2\]

    The product of the LCM and HCF of 6 and 8 is

    \[\text{LCM}(6,8)\times \text{HCF}(6,8)=24\times 2=48\]

    Similarly, the product of 6 and 8 yields 48. Thus, the formula holds.

    \[\text{LCM}(6,8)\times \text{HCF}(6,8)=6\times 8=48\]

    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.

    Find the LCM and HCF of \(\frac{2}{3}\) and \(\frac{4}{5}\).

    Solution

    The numerators here are 2 and 4 while the denominators are 3 and 5. Let us begin by finding the required components.

    \[\text{LCM of numerators}=\text{LCM}(2, 4)=4\]

    \[\text{LCM of denominators}=\text{LCM}(3, 5)=15\]

    \[\text{HCF of numerators}=\text{HCF}(2, 4)=2\]

    \[\text{HCF of denominators}=\text{HCF}(3, 5)=1\]

    Now that we have this information, we can find the LCM and HCF of the given fractions.

    \[LCM=\frac{4}{1}=4\]

    \[HCF=\frac{2}{15}\]

    Examples Involving LCM

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

    Use the prime factors method to deduce the LCM of 9, 21 and 36.

    Solution

    We shall start by breaking down each of these given numbers as a product of its prime factors.

    \(9=3\times 3\)

    \(21=3\times 7\)

    \(36=2\times 2\times 3\times 3\)

    We will now express these products in their exponent form.

    \(9=3^2\)

    \(21=3\times 7\)

    \(36=2^2\times 3^2\)

    The prime factors here are 2, 3 and 7 with 2, 1 and 1 being their highest power respectively. Therefore, the LCM is

    \(\text{LCM}(9, 21, 36)=2^2\times 3^2\times 7=252\)

    Here is another example.

    Use the common division method to find the LCM of 14, 54 and 77.

    Solution

    To solve this, we shall construct a division table and divide these given numbers by the smallest prime number until the final row has a quotient of all 1.

    214, 54, 77
    37, 27, 77
    37, 9, 77
    37, 3, 77
    77, 1, 77
    111, 1, 11
    1, 1, 1

    The LCM is the product of all the prime factors obtained in the first column of the table above.

    \(\text{LCM}(14, 54, 77)=2\times 3\times 3\times 3\times 7\times11=2\times 3^3\times 7\times 11=4158\)

    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.

    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?

    Solution

    Given the information above, we observe that Ella is able to divide her handful of sweets into groups of 3 and 4. The smallest possible number of sweets she could have would be the lowest common multiple of 3 and 4. Let us begin by listing the first few multiples of 3 and 4.

    Multiples of 3: 3, 6, 9, 12, 15,...

    Multiples of 4: 4, 8, 12, 16,...

    By looking at the lists above, we find that the LCM of 3 and 4 is 12. Since 12 is indeed divisible by 3 and 4 we can conclude that Ella has a total of 12 sweets.

    Thus, if Ella were to divide her sweets into 3 equal clusters, she would have 4 sweets in each pile. However, if Ella were to divide her sweets into 4 equal clusters, she would have 3 sweets in each pile.

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

    Emily has a backyard garden that can fit between 400 and 600 potted plants. One day, she came home with a number of potted plants that she would like to place in her garden. She is able to arrange these plants in rows of 63 and 81 pots. How many potted plants did she purchase?

    Solution

    To find the number of potted plants Emily has, we would need to find the lowest common multiple of 63 and 81 since she is able to arrange them in rows of 63 and 81 pots. Further note that given an interval restriction, we only need to list the multiples of 63 and 81 between 400 and 600.

    Let us now list down the multiples of 63 and 81.

    Multiples of 63: 441, 504, 567

    Multiples of 81: 405, 468, 567

    From here, we see that the LCM of 63 and 81 is 567. Thus, Emily purchased 567 potted plants for her backyard garden.

    Lowest Common Multiple - Key takeaways

    • The LCM is the smallest common multiple shared between a given set of numbers.
    • Properties of LCM
      • The LCM of two prime numbers is always the product of the given numbers.
      • The LCM of any set of numbers is never less than the given numbers.
    • Methods for finding the LCM
      1. Listing method;

      2. Prime factorisation method;

      3. Division method.

    Frequently Asked Questions about Lowest Common Multiple

    What are the rules of the Lowest Common Multiple? 

    • The LCM of two prime numbers is always the product of the given numbers. 
    • The LCM of any set of numbers is never less than the given numbers.

    What is Lowest Common Multiple?

    The LCM is the smallest common multiple shared between a given set of numbers. 

    What is an example of Lowest Common Multiple?

    • The LCM of 3 and 4 is 12
    • The LCM of 12, 24 and 48 is 48
    • The LCM of 7 13 is 91

    How to solve Lowest Common Multiple?

    There are 3 methods for finding the LCM: 

    1. Listing method;

    2. Prime factorisation method;

    3. Division method.

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