Factors and multiples are two key concepts studied together in math at the elementary level. Often students have a hard time knowing the difference between the two concepts. A scientific study reveals that the meaning students assign to the concepts of factors, divisors, and multiples is often different from the meaning assigned by mathematicians in the context of number theory, and that their links among the concepts are often weak and incomplete. If your child repeatedly makes mistakes while calculating factors and multiples of numbers, then it is time to revisit this concept. In this article, you will learn the relationship between factors and multiples, their definitions with examples, and the difference between them in detail.
What Is a Factor?
A factor is a number that divides a bigger number completely without leaving any remainder.
What Is a Multiple?
A multiple of a number is the resultant product when we multiply that number with another natural number. You can observe multiples in multiplication tables.
How to Find Out Factors/Multiples of a Given Number
To get a factor of a given number, you must determine another number that evenly divides it.
To get a multiple of a given number, you multiply that number by another integer.
Factors & Multiples – Word Problems
Factors and multiples can be best explained with the help of real-world examples.
Example 1: Billy wants to store 10 eggs in some baskets. How many baskets should Billy have so that the eggs are divided evenly and no egg is left out?
Method 1: Factors by division
Step 1: Find all the numbers less than or equal to the given number.
Step 2: Divide the given number by each of these smaller numbers. Pick those numbers that divide the given number completely. These divisors are the factors of the given number.
In this example, the numbers less than or equal to 10 are 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10. When we divide 10 by these numbers, we observe that only 2, 5, and, 10 divide the number 10 without any remainder.
Therefore, these divisors are the factors of the number 10.
Method 2: Factors by multiplication
Write the given number as the product of two numbers in as many possible ways as possible. The smaller numbers you get are the factors of the given number.
For this example, we will write 10 as the product of two numbers in multiple ways.
Thus, the factors of 10 are 1, 2, 5, and 10.
Conclusion: Billy should have 2, 5, or 10 baskets to divide the eggs evenly.
Example 2: Sally is preparing goodie bags for her friends. She has to put two candies in each bag. Are 5 candies enough to prepare three goodie bags?
Since we have 3 goodie bags, and we have to put 2 candies in each, we must find out the first three multiples of 2 to get the answer.
The first three multiples of 2 can be found by multiplying 2 by the first three natural numbers.
Conclusion: To prepare 3 goodie bags, we need 6 candies. Therefore, 5 candies will not be enough to prepare three goodie bags.
Factors vs Multiples
Following are the key differences between factors and multiples:
Factors | Multiples | |
1 | Factors consist of a list of numbers, each of which can divide a specific number without leaving a remainder. | Multiples are the resultant products of a number when we multiply that number with other natural numbers. |
2 | Factors of a number are limited. There is a minimum of two factors of every number, i.e. 1 and the number itself. | Multiples are unlimited. Every number is a multiple of 0 and itself. |
3 | Factors are generally less than or equal to the given number. | Multiples are greater than or equal to the given number. |
4 | We use division to get the factors of a number. | We use multiplication to get the multiples of a number. |
Fun Facts about Factors/Multiples
- Multiples of 2 will always be even numbers.
- The last digit of every multiple of 5 will always be 0 or 5.
- If the digits of a number add up to a multiple of 3, then the number is divisible by 3.
- If any natural number has only two factors (1 and the number itself), then the number is called a prime number. For example, 2 is a prime number with only two factors, i.e. 1 and 2.
- The factor of the number except 1 and the number itself is called a proper factor. For example, out of all the possible factors of 6 (1, 2, 3, 6), the proper factors are 2 and 3.
- There are ten multiples of 10 up to 100, and there are ten multiples of 100 up to 1000. Can you figure out the multiples of 1000 to get 10000?
Why Do Students Struggle with Factors/Multiples?
Some amateur math teachers fail to explain the concept of factors and multiples to their students simply because they don’t link these concepts to algebraic reasoning. These concepts become easy as pie when taught by an expert with the help of relevant tools and activities.
At Math Project, we publish informative math blogs regularly to help parents teach math to their kids at home. While teaching math at home can help your children master basic math concepts, it does not help you evaluate their progress as per international standards. Therefore, we strongly recommend that you seek professional help to assess your child’s math proficiency. Call now at 1-844-628-4243 and book a free assessment with us.
Citations
Connecting mathematics: Finding factors and multiples — http://doer.col.org
Factors, divisors and multiples: Exploring the web of students’ connections — https://www.researchgate.net
Linking Factors and Multiples to Algebraic Reasoning — https://doi.org/10.5951/MTLT.2019.0118
FAQs on Factors and Multiples
How are factors and multiples helpful in solving problems in the real world?
Factors and multiples are helpful in many ways. Understanding factoring allows you to easily navigate number relationships in the real world without relying on your calculator or phone to do the work for you. Here are some real-world applications of factors and multiples:
- Solving problems about the side lengths and areas of rectangles
- Exchanging money
- Comparing the price per unit for a group of similar products
- Calculating time slots
How are factors and multiples similar?
Factors and multiples are inverse concepts.
Which factors are multiples of 3?
Multiples of 3 are 3, 6, 9, 12, 15, 18, 21, and so on.
Multiples of 5 are 5, 10, 15, 20, 25, and so on.
Which factors of 42 are multiples of 6?
The factors of 42 are 1, 2, 3, 6, 7, 14, 21, and 42.
The multiples of 6 are 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, so on.
Filtering out the common numbers from the factors of 42 and the multiples of 6, we get 6 and 42.
Which factors of 48 are multiples of 4?
The factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48.
The multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, so on.
Filtering out the common numbers from the factors of 48 and the multiples of 4, we get 4, 8, 12, 16, 24, and 48.
What factors of 36 are multiples of 4?
The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36.
The multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, so on.
Numbers 4, 12, and 36 are factors of 36 that are also multiples of 4.
How many factors of 96 are multiples of 4?
The factors of 96 are 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96.
The multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, so on.
Filtering out the common numbers from the factors of 96 and the multiples of 4, we get 4, 8, 12, 16, 24, 32, and 48.
How many factors of 30240 are multiples of 105?
The factors of 30240 are 1, 2, 3, 5, 7, 135, 270, 540, 560, 864, 945, 1080, 1120, 1680, 1890, 2016, 2160, 3360, 3780, 4320, 5040, 6048, 7560, 10080, 15120, and 30240.
The multiples of 105 are 105, 210, 315, 420, 525, 630, 735, 840, 945, 1050 and so on.
Filtering out the common numbers from the factors of 30240 and the multiples of 105, we get 945. Therefore, 945 is the only number that is a factor of 30240 and a multiple of 105.
How many factors of 84 are also multiples of 4?
The factors of 84 are 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, and 84.
The multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, so on.
Filtering out the common numbers from the factors of 84 and the multiples of 4, we get 4, 12, and 28.
How many factors of 90 are also multiples of 6?
The factors of 90 are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, and 90.
The multiples of 6 are 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, so on.
Filtering out the common numbers from the factors of 90 and the multiples of 6, we get 6, 18, and 30.
How many factors of 100 are also multiples of 5?
The factors of 100 are 1, 2, 4, 5, 10, 20, 25, 50 and 100.
The multiples of 5 are 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, so on.
Filtering out the common numbers from the factors of 100 and the multiples of 5, we get 5, 10, 20, 25, 50, and 100.
How many factors of 360 are also multiples of 3?
The factors of 360 are 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360.
The multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, so on.
Filtering out the common numbers from the factors of 360 and the multiples of 3, we get 3, 6, 9, 12, 15, 18, 24, and 30.