Any concept of math is easier to understand when it’s done with the help of practical examples. After all, I have yet to see a student who could learn by blindly following theories! So let’s get into what is a multiple with this practical example.
Take a single-digit non-zero number for example the number 2 and add it any number of times you want to. Let’s go ahead and add the number 2, five times. We can write this as 2+2+2+2+2 and the answer to that problem would be 10. We will call this new number the product of 2 times 5.
Similarly, if we took the number 5, and added it two times, we would get 10. 10 is the product of 5 times 2.
The number 10 is a multiple of 2 (since 2 x 5 = 10) and is also a multiple of 5 (since 5 x 2 = 10).
If you are with us so far, then it’s going to be very easy for you to understand multiples and all its concepts which are included in this article. As mentioned before, there will be no shortage of examples and questions for you to practice along. The answer to each practice question will be given at the end of the article.
What is a Multiple?
To define it simply, a multiple of a number is the product of that number and another non-zero whole number.
For example, 12 is a product of 2 and 6. Both 2 and 6 are non-zero whole numbers.
When you keep finding the product of a number by multiplying it with a series of whole numbers in any numerical order, you will get a multiplication table of that number.
For this, we are going to continue with the number 7 and make up a multiplication table for it.
|7 х 1 = 7||7 х 6 = 42|
|7 х 2 = 14||7 х 7 = 49|
|7 х 3 = 21||7 х 8 = 56|
|7 х 4 = 28||7 х 9 = 63|
|7 х 5 = 35||7 х 1 0= 70|
Practice the same steps with the number 5.
The products to multiples of 5 are 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, when the table stops at 10.
Here, 7, 14, 21, 28, 35, 42, 49, 56, 63, 70 are all multiples of 7. We stopped the table at 10, but in reality, it can go up to infinity. You did a good job of understanding the basic concept of multiples. Let’s add a few rules or properties to make it even easier.
Properties of Multiple
- Any number that is a multiple of 2 is an even number for e.g., 2, 4, 6, 8…. etc
- All the non 2 multiple numbers are called odd numbers for e.g., 1, 3, 7, 9….. etc
- There can be infinite, i.e., no limit, number of multiples.
- Every integer is a multiple of itself.
- Smallest multiple is the number itself; in the above example, the smallest multiple was 7.
- A product is a multiple of all its factors.
- 0 is a multiple of anything.
- If a and b are multiples of x, then a + b and a – b are also multiples of x.
Factors Are Not Multiples
Often easily confused, factors are not like multiples at all. Essentially, factors are what we multiply to get a number, and multiple is the result of that multiplication when calculated with an integer.
While there are infinite numbers of multiples, factors are finite. They are also less than or equal to the integer or number, whereas a multiple of a number is always equal or greater than its numbers.
For Example 2×2=4, Here 2 is the factor of 4, and 4 is the multiple of 2.
Methods of Finding out Multiples
Least Common Multiple (LCM)
Finding out the least common multiple of any two or more numbers is a fun and sometimes necessary way of solving any problem. There are different ways of finding out the least common multiplies of two or more numbers. Here are the four most common ways:
- The listing method
- The tree factor method
- The ladder method
- LCM of mixed fraction through the ladder method
1. Listing Method
In the listing method, we find the least common multiple that exists in the listing of all the numbers available.
If we want to find out the LCM of 4 and 5 using the listing method, we have to write the multiple listing of both the numbers. This would go something like this:
The multiple listing of 4 is: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44…
The multiple listing of 5 is: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55…
Notice how both 20 and 40 are repeated in both the listings? These are the common multiples of 4 and 5, but we are looking for the least common multiple and that is 20.
As soon as you spot a common multiple in both the lists, you can stop the listing. It’s a quick way to spot the least common multiple. However, to do this, you would need to memorize all the multiples of a single number in a numeric, as shown above.
Find out the LCM of 8 and 10 using the listing method.
The LCM of 8 and 10 is 40.
2. Tree Factor
The tree factor, the factor tree, or prime factorization is a useful way of finding a factor of any integer until we have reached the prime number and cannot factor anymore.
Finding the least common multiple through the tree factor is a fun way of breaking down or decomposing a large number and building it back up. Through the tree factor method, you can conduct a prime factorization of any two or more numbers. It is a useful tool for breaking down large and difficult problems.
We are going to find the least common multiple of 18 and 12 using the tree factor method.
So we write down “18” and “12” separately and start writing down the factors of it beneath the numbers.
Here, for 18, the lowest factors are 2, 3, and 3. For the number 12, we have 2, 2, and 3. So we can write it like this:
18 = 2 x 3 x 3 12 = 2 x 2 x 3
Now here is the super important part so pay close attention. We need to list the prime numbers as many times as they occur most often for each of the numbers. For the number 18, we only have one “2” and two “3”. For the number 12, we have two “2” and only one “3”. So we will take the number 3 (two times) since that occurs the most often in the number 18, and we will take number 2 (two times) since that occurs the most often in the number 12.
Then multiply. So 3 x 3 x 2 x 2 = 36. 36 is the least common multiple of both 12 and 18.
Fun fact: knowing how to find the prime factors of any problem is especially useful if you want a career in encryption or cryptology.
Find out the LCM of 30 and 45 using the tree factor method.
The LCM of 30 ad 45 is 90.
4. Ladder Method
The ladder or grid is a method of finding out the least common multiple of any two or more numbers. You can find the LCM of larger numbers in a much faster way. The ladder or grid looks like a tic-tac-toe grid. It is a set of two horizontal parallel lines that intersect each other at a vertical angle.
You start writing the numbers on the top row and move down as you factor down each number until you reach a prime number.
For the ladder method, we are going to find the LCM of 18 and 12 again.
We have the prime numbers 2 and 3 on the vertical line and 3 and 2 on the horizontal line. These numbers are multiplied with each other to get the common multiple of both 18 and 12 that is 36.
2 x 3 x 3 x 2 = 36
As mentioned above, 36 is the multiple of 18 and 12.
Find out the LCM of 20 and 84 using the ladder method.
The LCM of 20 and 84 is 420.
4. LCM Of Mixed Fractions
The best way to find a common denominator of a proper or improper fraction is through the LCM method for mixed fractions. In this method, you use the ladder LCM method for both the denominator number of the fraction to come up with a common number.
If you want to add two fractions that have a different denominator, you will have to find a common multiple or a common denominator first. Assume that the denominators of two fractions are 8 and 12. You need to find out the lease common multiple between the numbers 8 and 12. Here the common multiple is 24. 8 times 3 is 24, and 12 times 2 is 24. However, 24 isn’t the only common multiple with both 8 and 12. We also have 72, i.e., 8 times 9 is 72, and 12 times 6 is 72.
So, what makes 24 special? It is the least common multiple of both denominators.
Grid Method Multiplication
The grid method, also known as the grammar school or box method, is a good place to start learning multiples and the calculations with larger numbers that are more than 10. This method breaks the process of manual multiplication into three steps.
Here we will find out the multiples of 15 with 37.
The first step is to make up a grid or table with the first column and first row being your numbers to be multiplied. The first box will remain empty. Like this:
The second step is to multiply all the numbers in the rows and the columns with each other. Like this:
The third step is a revelation of addition. When you add up 300 + 150 + 70 + 35, you will get a sum of 555. When you calculate 15 x 37, you will get exactly 555.
This method makes it easy to break down and simplify the factors of that number.
Calculate the product of 345 and 28 using the grid method.
The product 345 and 28 is 9660.
The product 345 and 28 is 9360.
This was a fun and informative part of math. We got to know what a multiple is, how we can multiply factors and the different methods of finding out the least common multiple. Multiples are incredibly fun and useful.