Factors / Factor Pairs
Basics on the topic Factors / Factor Pairs
Factors and Factor Pairs
Knowing how to find factors and factor pairs is very important because it helps you understand how numbers work together in multiplication and division.
Understanding Factors and Factor Pairs – Definition and Importance
A factor is a number that can be divided into another number evenly without leaving a remainder. Factor pairs are two numbers that, when multiplied together, result in a given product.
For example, in the number sentence:
2 x 4 = 8
Two and four are factor pairs of eight. Additionally, since multiplication is the inverse or opposite of division, we can say:
Factors of 8 = 1, 2, 4, 8
To find all the factor pairs of a number, a systematic approach, such as the "rainbow method," can be very helpful. The rainbow method uses counting numbers and tests for divisibility to identify factor pairs.
Factors and Factor Pairs – Example
Let's use the rainbow method to find the factors of eighteen.
- Start with the first counting number: 1. The factor pair for 1 is 18 because 1 x 18 = 18.
- Next, try 2. The factor pair for 2 is 9 because 2 x 9 = 18.
- Now, try 3. The factor pair for 3 is 6 because 3 x 6 = 18.
- Try 4 and 5. Neither can be multiplied by any whole number to give 18.
- Notice that 6 has already been paired with 3, which means we have found all the factor pairs for 18.
So, the factors of eighteen are 1, 2, 3, 6, 9, and 18.
Factors and Factor Pairs – Guided Practice
Let's find the factor pairs of the number forty together.
Factors and Factor Pairs – Application
Now, find the factor pairs and list the factors of seventy-eight on your own.
Factors and Factor Pairs – Summary
Key Learnings from this Text:
- A factor is a number that divides into another number evenly without a remainder.
- Factor pairs are two numbers that, when multiplied together, give a specific product.
- The "rainbow method" is a systematic approach to finding all factor pairs of a number.
- Example factors of 18 are 1, 2, 3, 6, 9, 18.
- Example factors of 40 are 1, 2, 4, 5, 8, 10, 20, 40.
- Example factors of 78 are 1, 2, 3, 6, 13, 26, 39, 78.
Explore more content on our website, such as interactive practice problems, videos, and printable worksheets, to support your educational journey.
Factors and Factor Pairs – Frequently Asked Questions
Transcript Factors / Factor Pairs
Skylar and Henry found a pot of gold at the end of a rainbow, but it is guarded by a cunning troll. He demands that they find all the numbers that divide into seventy-eight evenly before they can take it. But first, Skylar and Henry must learn about factors and factor pairs. A factor is a number that can be divided into a whole number evenly. Because multiplication is the inverse of division, we can say that a factor multiplied by another factor equals a product. Factor pairs are the two numbers we multiply together to get the product. In this number sentence, the factor pair of eight is two and four. But these are not the only factor pairs that equal eight. We know that we can get any whole number by multiplying one by the number itself, so one and eight are also a factor pair of eight. From our sets of factor pairs, we can list the factors for a number. The factors of eight are one, two, four, and eight. When finding all the factor pairs of a number, it helps to use a system. One system is called the "rainbow method." The "rainbow method" uses counting numbers and divisibility. Divisibility means it can be divided evenly by another number. Let's use this system to find the factors of eighteen. To start, put the first counting number, "one," and the partner factor, "eighteen," here. Now, write the next counting number, two... and think: what number do we multiply two by to make eighteen? (...) Nine. What about three? (...) Three times six makes eighteen. Now, try four. Can we multiply four by a number to get eighteen? (...) No, four is not a factor of eighteen. Neither is "five." We can see that "six" is already paired on this side of the rainbow, so we have found all the factor pairs. Now we list them in order. The factors of eighteen are one, two, three, six, nine, and eighteen. Let’s try the number forty. We have one and (...) forty,... and two and (...)twenty. What is the next number that would be a factor of forty? (...) Four. What is the partner factor for four? (...) Ten. What is the next set of factor pairs? (...) five and (...) eight. Is six a factor of forty? (...) No. How about seven? (...) No, forty is not divisible by either number. Eight is already on this side of the chart, ... so we have found all of the factor pairs for forty. Now, list the factors in order… One, two, four, five, eight, ten, twenty, and forty Here’s one to try on your own. Find all the factor pairs of seventy-eight and then list the factors in order. Remember, to find the pairs, go through all the counting numbers and check for divisibility. Pause the video to solve, and press play when you are ready to continue. The factor pairs of seventy-eight are... one and (...) seventy-eight; two and (...) thirty-nine; three and (...) twenty-six; and six and (...) thirteen. The factor pairs of seventy-eight are... and we list all the factors in order like this. Remember, a factor is a number that can be divided into a whole number evenly. Because multiplication is the inverse of division, we can say that a factor multiplied by another factor equals a product. Factor pairs are the two numbers we multiply together to get the product. And with that, the troll gave a wave and disappeared. “We’re RICH!” “It’s (...) it’s (...) it’s CHOCOLATE!” “Mmmm, rich dark chocolate!”
Factors / Factor Pairs exercise
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Find the factors of 6.
HintsWhat number do we multiply 2 by to make 6?
2$\times$ ? = 6
SolutionUsing the Rainbow Method, we start with the first counting number: 1. What number do we multiply 1 by to make 6? 6. Therefore, the first factor pair is (1, 6).
The next counting number is 2. What number do we multiply 2 by to make 6? 3. Therefore, the next factor pair is (2, 3).
3 is the next counting number, but 3 is already one of the factors on the right. Therefore, there are no more factor pairs.
The factors of 6 are: 1, 2, 3, and 6. The missing factor in this problem is 3.
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Which of the following are factor pairs of 28?
HintsWhat number do we multiply 1 by to make 28?
What number do we multiply 2 by to make 28?
What number do we multiply 4 by to make 28?
SolutionThe factor pairs of 28 are:
- (1, 28)
- (2, 14)
- (4, 7)
Using the Rainbow Method, we start with the first counting number: 1. What number do we multiply 1 by to make 28? 28. Therefore, the first factor pair is (1, 28).
The next counting number is 2. We multiply 2 by 14 to make 28, so the next factor pair is (2, 14).
3 is not a factor of 28 because dividing 28 by 3 does not give us a whole number. However, 4 is a factor of 28: 4$\times$7 = 28. Therefore, (4, 7) is also a factor pair of 28.
5 and 6 are not factors of 28. 7 is a factor, but is already one of the factors listed.
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Find the factors.
HintsUsing the Rainbow Method, we start with the first counting number: 1. What number do we multiply 1 by to make 50?
What number do we multiply 2 by to make 50?
What number do we multiply 5 by to make 50?
SolutionUsing the Rainbow Method, we start with the first counting number: 1. What number do we multiply 1 by to make 50? 50. Therefore, the first factor pair is (1, 50).
The next counting number is 2. We multiply 2 by 25 to make 50, so the next factor pair is (2, 25).
3 is not a factor of 50 because dividing 50 by 3 does not give us a whole number. Neither is 4. 5 is a factor of 50, however, so the next factor pair is (5, 10).
6, 7, 8, and 9 are not factors of 50. 10 is a factor, but is already one of the factors listed.
Therefore, the factors of 50 are: 1, 2, 5, 10, 25, and 50.
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Find factor pairs.
HintsEach product has 2 corresponding factor pairs.
Try using the Rainbow Method.
We can get any whole number by multiplying 1 by the number itself.
For example: the product 7 has factors 1 and 7.
If we multiply the factors in a factor pair by each other, we get the product.
For example, 6 has factors (2, 3). 2$\times$3 = 6.
Solution- (1, 10) and (2, 5) are factor pairs of 10.
- (2, 28) and (7, 8) are factor pairs of 56.
- (1, 38) and (2, 19) are factor pairs of 38.
- (3, 15) and (5, 9) are factor pairs of 45.
To find the factor pairs of 45:
- You can use the Rainbow Method. We start with the first counting number: 1. The first factor pair is (1, 45). 45 is not divisible by 2, so the next counting number is 3. We multiply 3 by 15 to make 45, so the next factor pair is (3, 15). The next counting number is 5, so the next factor pair is (5, 9). This is one of the available elements, so (5, 9) should be matched with 45.
- Another method to find the corresponding product is to multiply the two factors in each factor pair together. For example: 5$\times$9 = 45.
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Find the factors of 12.
HintsWhat number do we multiply 2 by to make 12?
2$\times$ ? = 12
SolutionIf we use the Rainbow Method, we can see that 2 is the only factor listed without a partner factor. What number do we multiply 2 by to make 12? 6.
The missing factor of 12 is 6.
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Find all of the factor pairs of 84.
HintsOne method to find the partner factor is to divide the product by the factor given.
For example: 84$\div$2 = 42. Therefore, the partner factor for 2 is 42.
Can you use the Rainbow Method to find all of the remaining factor pairs?
SolutionThe factor pairs of 84 in order of counting numbers are: (1, 84), (2, 42), (3, 28), (4, 21), (6, 14), and (7, 12).
One method to find each partner factor is to use the Rainbow Method to find all of the factor pairs of 84.
Another method is to divide the product by the factor given.
For example: 84$\div$2 = 42. Therefore, the partner factor for 2 is 42.
