Joseph has a master's degree in literature as well as alternative teaching and ESL educator certifications. He has worked with middle school, high school, and college students in writing and language arts.
Factor Trees and Factors
When we make a family tree, we make a chart of the people that came before us. They're called family trees because they tend to branch out as they progress, as each ancestor gives way to more ancestors. In the same way, a factor tree shows the numbers that come together to create a larger number, and it branches out as it progresses because each resulting factor then gives way to its own factors. In other words, a factor tree is a tool that breaks down any number into its prime factors.
Before we go into detail about factor trees, we need to review factors. Factors are simply numbers you can multiply together to get a certain product. For example, if we're looking at 2 * 3 = 6, then 2 and 3 are factors of 6. But, we could also say that 1 * 6 = 6. So 6 has four factors: 1, 2, 3, and 6. This factor business could get messy with large numbers because, generally speaking, the larger a number is the more factors it will have. That's where the factor tree comes in.
For an example of a factor tree, let's start with 100. We can split that number into any of its two factors. In other words, we're going to find two numbers that, when multiplied together, give us a product of 100. Let's start with 4 and 25 because 4 * 25 = 100. And, just like we would in a family tree, we're now going to find the 'ancestors,' or factors, of 4 and 25. We'll start with 4, which we could write as 2 * 2, so we'll put those factors into the tree under 4.
Now, if we were to break down 2 into its factors, we'd get 2 and 1. Then, we could break down 2 into 2 and 1 again. This could go on infinitely because 1 and 2 are 2's only factors. That makes 2 a prime number, a number whose only factors are 1 and itself. So instead of having an endless chain of twos and ones on the factor tree, we'll stop at 2. In relation to the factor tree, we call the prime numbers prime factors because when all of them are multiplied together they give the original number. We always stop a factor tree's branch at the prime factors to avoid an infinite chain of the same factors.
So we're done with 4's branches. Let's look at 25. We could write that number as 5 * 5, so we'll place those factors on the tree now. Just like 2, 5 is a prime number because its only factors are 1 and 5. Since it's a prime number, we'll end those branches. The factor tree doesn't necessarily produce all of a number's possible factors. For example, 10 is a factor of 100 (10 * 10), but it doesn't show up on the tree.
What the tree does give us is a number's prime factorization. A number's prime factorization is simply the list of prime numbers you would multiply together to get a certain product. On a factor tree, all the prime numbers at the ends of branches give us the prime factors. So 100's prime factorization is 2 * 2 * 5 * 5, or 2^2 * 5^2. Every number has one unique prime factorization, just like every person has one unique fingerprint.
Many Trees, One Prime Factorization
Unlike a family tree, which is fixed and can only occur in one way, some factor trees have different possible arrangements. Let's reconsider 100's factor tree. We could have started with 10 * 10 because, just like 25 * 4, it also gives us a product of 100. Let's set up another factor tree for 100 using 10 and 10 as the first pair of factors.
Now, we'll break down 10. We could write 10 as 5 * 2, so those will be our next factors in the tree. And, since 2 and 5 are prime numbers, we'll end the branches there. Notice that even though we started with a different pair of factors for 100, we still end up at the same prime factorization: 2^2 * 5^2. No matter how we start a factor tree, if completed correctly, it will always reveal a number's prime factorization.
A Large Factor Tree
Now that we've seen a factor tree in action, let's challenge ourselves with a larger number. We'll use a factor tree to find the prime factorization of 2,310. Even though 2,310 is a much larger number than 100, we'll still follow the same steps to create its factor tree. In fact, we'll start with 231 and 10 as the first pair of factors.
We can write 231 as 77 * 3. We can also break 10 down into its factors, 2 and 5. Now we have the prime factors of 2, 3, and 5. But, we can still break 77 down into 7 * 11. Now, all our branches end in prime numbers. So 2,310's prime factorization is 2 * 3 * 5 * 7 * 11. We could have started this tree with any factor pair, such as 1,155 * 2 or 770 * 3. When we carried out the factor tree, we would still discover the same prime factorization.
- Factors are the numbers you multiply together to get a certain product.
- A prime number is a number whose only factors are 1 and itself.
- A factor tree is a tool that breaks down any number into its prime factors.
- A certain number's prime factorization is the list of prime numbers or prime factors that you would multiply together to create that certain number.
- No matter how you start a number's factor tree, it will always give you the one unique prime factorization for that number.
Factor tree: a tool that breaks down any number into its prime factors
Factors: numbers you can multiply together to get a certain product
Prime number: a number whose only factors are 1 and itself (also called prime factors)
Prime factorization: the list of prime numbers you would multiply together to get a certain product
Utilize this lesson's main points as you prepare to:
- Distinguish between factors and a factor tree
- Understand the process of splitting numbers
- Use a factor tree
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