How to Build and Use a Factor Tree in Prime Factorization
The diagram used to calculate the prime factors of a natural number greater than one is a Factor Tree.
Factor Tree Method
Basic steps in the Factor Tree Method as follows:
In constructing a Factor Tree, the first step is to find a pair of factors whose product is the number we are factoring in. The first branch in the Factor Tree is these two variables.
There are often several distinct pairs of variables that we might choose to start the process. Here we can start with any two variables.
For each factor, we repeat the process until every tree branch ends in a primary. The prime factorization is done then.
The Fundamental Theorem of Arithmetic ensures that the same, unique prime factorization for the number can result in all prime factorizations of the same number.
For example, the number 30 can be written as 2 x 3 x 5 in prime factorization which is found by Factor Tree method as follows:
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Prime Factorization By Factor Tree Method
Prime factorization is a method of factoring a number in terms of prime numbers, i.e. finding a number's prime factors, so that these factors will equally divide the original number.
The Prime Factorization will use Division method or Factor Tree method to find the Prime factors of any numbers.
The logic behind Prime factorization is to divide the given number by prime factors until we get the remainder as equal to 1.
36 Factor Tree
The Prime Factor Tree for 36 is given as below:
Step 1: Divide 36 by the lowest prime number. Here we are dividing by 2.
Step 2: After dividing 36 by 2 we get 18, divide again by the lowest prime number. Here we will divide by 2 again.
Step 3: After dividing 18 by 2 we get 9, the lowest prime number which divides 9 is 3. So we get 3 and dividing it again by 3 we will get the remainder as 1. So this will give us the Prime Factorization by Factor Tree method.
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The prime factorization of 36 from the Prime FactorizationTree Method is 2 x 2 x 3 x 3.
Factor Tree of 65
The Prime Factor Tree for 65 is given as below:
Step 1: Divide 65 by the lowest prime factor. Here 65 is divisible by 5 which will give 13.
Step 2: So 13 is only divisible by 13 which will give the remainder as 1. So this will give us the Prime Factorization by Factor Tree method.
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The prime factorization of 65 from the Prime Factorization Tree Method is 5 x 13.
Factor Tree Method of 48
The Prime Factor Tree for 48 is given as below:
Step 1: Divide the number 48 by the least possible prime factor. Here 48 is divisible by 2 which gives 24.
Step 2: Divide 24 again by 2 which will give us 12.
Step 3: Divide 12 by 2 which will give us 6.
Step 4: Dividing 6 by 2 we will get the remainder as 3.
Step 5: Since we have to get the final remainder as 1 divide 3 by 3. So this will give us the Prime Factorization by Factor Tree method.
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The prime factorization of 48 from the Prime Factorization Tree Method is 2 x 2 x 2 x 2 x 3.
Conclusion
We can conclude by saying that the Factor Tree is a diagrammatic method used to determine the prime factors of any natural number greater than one. This is one of the simplest and easiest methods to find Prime factorization of any numbers.
FAQs on Factor Tree Explained for Students
1. What is a factor tree in Maths?
A factor tree is a visual diagram used in mathematics to break down a composite number into its prime factors. It starts with the original number at the top and branches out into pairs of factors. This process continues, breaking down each composite factor until all the branches end in prime numbers.
2. How do you create a factor tree for a number, for example, 36?
To create a factor tree for 36, you follow a simple process of division until you reach prime numbers. Here are the steps:
- Start by writing 36 at the top of your page.
- Break 36 into a pair of factors, such as 6 and 6, and draw branches to them.
- Since 6 is not a prime number, break each 6 down further into its factors, which are 2 and 3.
- Circle the numbers at the end of the branches (2, 3, 2, 3). These are all prime numbers, so the tree is complete.
- The prime factorization of 36 is 2 × 2 × 3 × 3.
3. What is the main purpose of using a factor tree?
The primary purpose of a factor tree is to find the prime factorization of a number. This means expressing a composite number as a unique product of its prime factors. This skill is a fundamental building block in number theory and is essential for more advanced calculations, such as simplifying fractions and finding the Highest Common Factor (HCF) and Lowest Common Multiple (LCM) of numbers.
4. Why are only prime numbers found at the end of a factor tree's branches?
A factor tree is a process of decomposition. You continue breaking down numbers into smaller factors until you can't go any further. A prime number is a whole number greater than 1 that cannot be formed by multiplying two smaller whole numbers. Therefore, when a branch reaches a prime number, it is a natural endpoint because it cannot be factored any further. The goal of the factor tree is to find these fundamental, indivisible building blocks.
5. Can a single number have more than one unique factor tree?
A number can have different-looking factor trees depending on the initial pair of factors you choose. For example, a factor tree for 24 could start with 4 × 6 or 2 × 12. However, the final set of prime factors at the ends of the branches will always be the same (three 2s and one 3). This principle is known as the Fundamental Theorem of Arithmetic, which states that every integer greater than 1 has a unique prime factorization.
6. How is a factor tree useful for finding the HCF and LCM of two numbers?
A factor tree provides the prime factors needed to calculate the HCF and LCM efficiently. After creating a factor tree for each number:
- To find the HCF (Highest Common Factor), you identify all the prime factors that are common to both trees and multiply them together.
- To find the LCM (Lowest Common Multiple), you multiply the highest power of all prime factors that appear in either tree.
7. What is the difference between a 'factor' and a 'prime factor' in a factor tree?
In the context of a factor tree, a 'factor' can be any number along the branches that divides the original number, including composite numbers (like 4, 6, or 12). These are the intermediate steps in the process. A 'prime factor' is a specific type of factor that is also a prime number. These are found only at the very ends of the branches, as they represent the final, indivisible components of the original number.
