The lowest common multiple (LCM) of given numbers is the lowest or least multiple that the given numbers have in common and it is exactly divisible by all the given numbers. The lowest common multiple is also referred to as the least common multiple (LCM).
How to find the lowest common multiple of numbers?
In finding the lowest common multiple (LCM) or least common multiple, you can use these two methods which will be discussed in this article.
• Using multiples
• Using product of primes
Steps in finding the lowest common multiple of a number
To find the Lowest common multiple using the method of multiples;
• list all the multiples of the given numbers
• pick out all the common multiples among the multiples of the given numbers.
• The lowest multiple among the common multiples is the lowest common multiple (LCM) of the numbers.
Before we look at some examples of this method, let’s try to know how to find the multiples of numbers.
Multiples of a number
multiple of a given number is found by multiplying the number with other whole numbers. For instance, 5 × 8 = 40. In this case, 40 is a multiple of 5 and also a multiple of 8.
In general, when a whole number is exactly divisible by another whole number, then the first number is a multiple of the second number. For example, 12 is divisible by 3. Hence 12 is a multiple of 3.
Example 1: find the LCM of 3 and 4
answer: as we discussed in the steps above list all the multiples of 3 and 4
Multiples of 3 = {3,6,9,12,15,18,21,24,27,30,33,36…}
Multiples of 4 = {4,8,12,16,20,24,28,32,36,40,44,48…}
Pick out the common multiples of 3 and 4
common multiples of 3 and 4 = {12,24,36}. In this case, 12 is the least or smallest multiple among the common multiples. Hence the LCM of 3 and 4 is 12.
That’s pretty cool, right? Let’s look at another example of this method.
Example 2: find the LCM of 9, 15, and 18.
Answer: list all the multiples of 9, 15 and 18
Multiples of 9 = {9,18,27,36,45,54,63,72,81,90,99…}
Multiples of 15 = {15,30,45,60,75,90,105…}
Multiples of 18 = {18,36,54,72,90,108…}
The common multiples of 9, 15, and 18 = {90}. Therefore the LCM of 9, 15, and 18 is 90.
This is pretty good and simple. Now let’s turn our attention to the second method (product of primes)
Steps in finding the Lowest common multiple using the product of primes.
To find the Lowest common multiple (LCM) of given numbers using the product of primes follow the steps
• Write down each of the given numbers as a product of primes (index notation form)
• pick out the highest power of each prime number that has occurred and multiply them all together.
This will be clearly explained in the examples here.
Example 1. Find the LCM of 12 and 15
Answer: write the numbers given as a product of primes
\begin{aligned} &12 = 2\times 2\times3 =2^2\times3\\
&15 = 3\times5 \end{aligned}\\
\text{Therefore, the LCM of 12 and 15 } = 2^2\times3\times5=4\times 3\times5 = 60
This is so simple, right? Let’s run over another example under this method.
Example 2. Find the LCM of 12, 28, and 45
Answer: write down the given numbers as a product of primes
\begin{aligned}&12 = 2\times2\times3 = 2^2\times3\\
&28 = 2\times2\times7=2^2\times7\\
&45=3\times3\times5=3^2\times5\\\end{aligned}\\
\text{Therefore, the LCM of 12, 28 and 45} = 2^2\times3^2\times5\times7= 4\times9\times5\times7=1260
Solving practical problems involving Lowest common multiple (LCM)
The lowest common multiple (LCM) can be used to solve practical problems. The problem will not ask you to “find the LCM of”. There are clues to look out to. Let’s discuss the clues
• when the question says something like “repeating pattern of the numbers”. This clue means to find the Multiples.
• Then again when the question says something “to find the first time the match after the start” this clue means find the LCM.
Let’s go ahead to look at some example
Example 1. Esther belongs to three committees in a high school. One of the committees meets every 4 days, another committee meets every 5 days and the third Committee meets every 6 days. They all meet on the first day of the school term. How many days after this will they meet on the same day?
Answer: This question is talking about the LCM of 4,5 and 6. Using a product of primes, write down the given numbers as a product of primes
\begin{aligned}&4 =2×2= 2^2\\
&5= 5\\
&6= 2×3\\\end{aligned}\\
\text{Therefore, the LCM of 4 5 and 6} = 2^2×3×5=4×3×5=60\\
\text{Hence,the three committees will together on the 60th day.}
