In this article, we cover the world of multiples. The main focus, however, is on what the lowest common multiple is. All the information you need to understand this concept is provided in this article. We also provide examples to help you master arithmetic quickly!
What is a multiple?
A multiple is a number that may be divided into two parts without leaving a remainder. It can be helpful for children to think of it as a number in another number’s timetable. For example, 24 is a multiple of 12 as well as 1, 2, 3, 4, 6, 8, and 24. Factors and multiples are linked. For example, 4 is a factor of 12 and 12 is a multiple of 4.
What is a common multiple?
The Least Common Multiple (LCM) is also known as the Least Common Divisor (LCD). The LCM is the smallest positive integer that is evenly divisible by both a and b for two integers, abbreviated LCM (a,b). LCM(2,3), for example, equals 6 and LCM(6,10), equals 30.
The least common multiple (LCM) of two or more numbers is the smallest number equally divisible by all of the numbers in the set.
What is the point of LCMs?
Lowest common multiples are useful when adding or subtracting fractions, or comparing fractions in the same denomination. For example, to calculate 3/5 + 1/6, you need to calculate the lowest common multiple of 5 and 6 to determine the common denominator (30). The fractions can then be converted to 18/30 + 5/30 = 23/30.
What is the easiest way to find common multiples?
To determine the common multiples of a group of numbers, you must first list all of the numbers’ multiples and then begin selecting the common multiples.
By listing their multiples, you may quickly find the frequent multiples of two numbers. You can indicate or circle the multiples that are shared by both numbers after listing the multiples of the specified integers. These are the two numbers’ common multiples.
Let’s look for typical multiples of 2 and 5 as an example:
- 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, and so on are all multiples of two.
- 5, 10, 15, 20, 25, 30,… are all multiples of five.
As a result, popular multiples of 2 and 5 include: 10, 20,… and so on. It’s worth noting that all of the common multiples are divisible and can be divided by either 2 or 5.
LCM of a Set
You can identify the common multiples of three numbers by using the same technique you used to find the common multiples of two numbers. The common multiples are the multiples that the three numbers have in common.
Let’s see what the most common multiples of 10, 20, and 30 are. There are several multiples that are prevalent in the multiples of 10 and 20, but not in the multiples of 30. As a result, they cannot be regarded as common multiples of all three numbers. You must choose the multiples that are common to all three values.
To find their common multiples, make a list of the multiples of 10, 20, and 30.
- 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120, and so on are all multiples of ten.
- 20, 40, 60, 80, 100, 120, 140, 160, 180, and so on are all multiples of 20.
- 30, 60, 90, 120, 150, 180, 210, and so on are all multiples of 30.
60 and 120 are typical multiples of 10, 20, and 30, and so on.
What are the properties of the least common multiple (LCM)?
The lowest common multiple is the smallest number that can be divided by the provided numbers. Various approaches, such as the listing method, prime factorization method, and division method, can be used to calculate the least common multiple (LCM) of numbers.
Here are the properties of an LCM:
- The least common multiple (LCM) of two or more numbers cannot be smaller than one of them. The LCM of 3, 8, and 12 equals 24, which is not less than any of the given values.
- A number’s LCM is the greater number itself if it is a factor of another number. The LCM of 8 and 16 is, for example, the number 16 itself.
How to find the least common multiple of a number?
Common multiples method
For this method, list each number’s multiples until at least one of them appears on all of the lists. Then, on all of the lists, find the lowest common number. This is the LCM!
For example: LCM(6,7,21)
- Let us firstly list the multiples of 6:
6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 72,
- Next, let’s do the same for 7:
7, 14, 21, 28, 35, 42, 56, 63
- Lastly, we need to list the multiples of 21:
21, 42, 62…
Find the least number that appears on each list. It’s highlighted in bold above.
As a result, LCM(6, 7, 21) equals 42.
Here is another example. Let’s use the listing approach to find the LCM of 3 and 7.
- 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42,… are all multiples of three.
- 7, 14, 21, 28, 35, 42, 49, and so on are all multiples of seven.
The typical multiples of 3 and 7 are 21, 42, and so on, as can be seen. The smallest multiple among these common multiples is 21. Because it is the smallest of all the common multiples, the LCM of 3 and 7 is 21. As a result, the LCM of 3 and 7 equals 21.
Prime Factorization
For this method you have to determine all of a number’s prime factors. List all prime numbers found, in the order in which they appear most frequently for each given number. In order to find the lowest common multiple, write the list of prime factors and multiply them.
Finding the prime factorization of both a and b yields the LCM(a,b). Use the same procedure to find the LCM for more than two numbers.
For example, for LCM(12,30) we find:
- Prime factors of 12 = 2, 2, 3
- Prime factors of 30 = 2, 3, 5
We take the sum of all prime numbers found in the order in which they occur most frequently: 2 × 2 × 3 × 5 = 60
Therefore, LCM(12,30) = 60
Greatest Common Factor
For this method, we use the greatest common factor method. Firstly, what is a factor? A factor is a number that is produced when two numbers are divided evenly. A factor is also known as a divisor in this context. The highest number shared by all the factors is the greatest common factor of two or more numbers.
The formula for calculating the LCM of a collection of numbers using the greatest common factor (GCF) is: LCM(a,b) = (a×b)/GCF(a,b)
Here is an example. Find LCM(6,10)
- Factors of 6: 1, 2, 3, 6
- Factors of 10: 1, 2, 5, 10
GCF(6,10) = 2
Calculate (6×10)/2 = 60/2 = 30 using the LCM by GCF algorithm.
As a result, LCM(6,10) = 30.
Cake Method
Another method is the cake method which uses division to find the LCM. Because it is a simple division, people consider the cake or ladder approach to be the quickest and easiest way to find the LCM.
The cake method is also known as the ladder method, multiplication method, box method, factor box method, or the grid method of shortcuts. The boxes and grids may differ in appearance, but they always use prime division to find LCM.
Find the LCM (10, 12, 15, 75)
- Make a cake layer with your numbers (row)
10 12 15 75
- Divide the layer numbers by a prime number that is equally divisible by two or more of the layer’s numbers, then bring the result down to the next layer.
2: 10 12 15 75
5 6 – –
- If any integer in the layer is not equally divisible, simply bring it down.
2: 10 12 15 75
5 6 15 75
- Continue splitting the layers of the cake into prime numbers. You’re done when there are no more primes that can be evenly divided into two or more numbers.
2: 10 12 15 75
3: 5 6 15 75
5: 5 2 5 25
1 2 1 5
The LCM is the product of primes in the left column and bottom row (bold)
LCM = 2 x 3 x 5 x 2 x 5
= 300
The lowest common multiple of 10, 12, 15, and 75 is therefore 300!
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