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What is a Factor Tree? - Definition & Example

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  • 0:01 Factor Trees & Factors
  • 1:06 Splitting Numbers
  • 1:38 Prime Factors
  • 3:26 Many Trees, One Prime…
  • 4:20 A Large Factor Tree
  • 5:28 Lesson Summary
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Lesson Transcript
Instructor: Joseph Vigil
In this lesson, you'll review factors and prime numbers, and also learn what a factor tree is and see how it works to reveal any number's prime factorization. Then, you can test your new knowledge with a brief quiz.

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.

Splitting Numbers

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.

Prime Factors

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.

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