How to find the gcf using factor trees


Use Prime Factorization to Find GCF

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In my other lesson, I discussed the procedure on how to find the Greatest Common Factor using the List Method. This method is only effective when dealing with smaller numbers.

That’s why we need to learn a backup method to determine the GCF when larger numbers are involved. This alternate method takes advantage of the usefulness of Prime Factorization.

I must caution you that there is a prerequisite for this lesson. You will need to have an understanding of how to perform prime factorization on an integer.

Please don’t feel bad if you need to take the refresher lesson. Trust me, it is easy! You will soon realize that you are back here in no time. Here’s the link: Integer Prime Factorization


Steps on How to Determine the GCF using Prime Factorization

These are the steps on how to find the greatest common factor of two numbers using Prime Factorization. Although this method can be extended to find the GCF of multiple numbers, I just want to focus on two numbers. 2} and 5.

For the last step, we multiply together the numbers we have chosen from the previous step to determine the GCF. Thus, the GCF of 150 and 180 is 2 times 3 times 5 which is equal to 30.


Example 3: What is the GCF of 1,260 and 1,960?

Let’s take this skill of finding the greatest common factor to the next level. This time we will find the GCF of two numbers that have numerical values between 1,000 and 2,000.

By now, you’ve probably realized that finding the GCF of these numbers using the list method is going to be cumbersome. This is the reason why I have to create a separate lesson on finding the GCF using the Prime Factorization method. It is more efficient, accurate, and prone to less errors especially when you are working it out by hand.

To get to the heart of this method, you will need to have a good grasp on how to prime factorize a positive integer using the Prime Factor Tree.

I can’t overemphasize the importance of Prime Factor Tree. Trust me, it will be your best friend from here on out during your study of algebra in general.

In a nutshell, here’s the method of prime factorization using the Prime Factor Tree. Start dividing the given number by the smallest prime number which is 2. If 2 evenly divides the number, draw a diagonal down towards the left (branch of the tree) of the given number and write 2. Then, write the quotient by drawing a diagonal down towards the right. The quotient will become part of the trunk of the tree.

Keep dividing the subsequent quotients by 2 while recording your results as branches (divisors) and part of the trunk (quotients that are composite numbers). Keep going until such time when 2 can no longer divide the quotient. That’s when you move to the next prime number which is 3, and so on. Repeat the process until the quotient is a prime number. This is when you stop.

❖ Here’s the Prime Factor Tree of the number 1,260 and its Prime Factorization. 2} times 5 times 7 which gives us 140.


You might also be interested in:

Finding GCF using the List Method

Finding LCM using the List Method

Use Prime Factorization to Find LCM

Factor Tree - Method, Examples, FAQs

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A factor tree is created to find the prime factors of a number. It is created in the form of a tree in which the given number is split into branches that represent all the factors of the number. Let us learn more about finding the factors of a number using the factor tree method.

1. What is a Factor Tree?
2. How to Draw a Factor Tree?
3. Factor Tree of 36
4. Factor Tree of 48
5. FAQs on Factor Tree

What is a Factor Tree?

A factor tree is a method of factorizing a number that is similar to the way in which the branches of a tree are split. Every branch of the factor tree is split into factors and once the factors cannot be factorized further, the branches come to an end, the final factors are circled and are considered to be the prime factors of the given number.

How to Draw a Factor Tree?

A factor tree can be drawn by factorizing a number until we reach its prime factors. These factors are split and written in the form of a tree. Let us see how to draw the factor tree of 24.

Factor Tree of 24

The prime factorization of a number can be done by using the factor tree method. This is done by drawing the branches of the given number and writing the factors at the end of each branch. Once we reach that number which cannot be split any further, we stop and the factor tree is complete because all its prime factors are listed. At this stage, all the prime numbers that were circled are listed as the prime factors of the given number. Let us understand this by finding the factors of 24 using the factor tree method.

The factors of 24 can be calculated using the following steps. Observe the following figure to see the factor tree of 24.

  • Step 1: Find two factors of 24. We get 4 and 6.
  • Step 2: Observe these factors to see if they are prime or not.
  • Step 3: Since both 4 and 6 are composite numbers, they can be further split into more factors. Hence, we repeat the process of factorizing them and splitting them into branches until we reach the prime numbers.
  • Step 4: Here, 4 can be further split into 2 and 2. Similarly, 6 can be further split into 2 and 3. At this stage, we reach the prime numbers, 2 and 3. We circle them since we know that they cannot be factorized further. This is the end of the factor tree.
  • Step 5: Finally, we list all the circled factors as the prime factors of 24. This shows that the prime factors of 24 = 2 × 2 × 2 × 3

Factor Tree of 36

The factor tree of 36 can be drawn using the following steps.

  • Step 1: We start factorizing 36 and we split it into 2 and 18.
  • Step 2: Since 2 is a prime number, we cannot factorize it further, so, we circle it and we will come back to it later. However, 18 can be split further into 2 and 9.
  • Step 3: We know that 2 is a prime number so we circle this 2 since it cannot be split further. But, we can factorize 9 into 3 and 3.
  • Step 4: After factorizing 9 into 3 and 3, we finally reach the stage when all the factors are prime numbers. Hence, we stop and the factor tree of 36 is complete.
  • Step 5: Now, we list all the circled factors as the prime factors of 36. This shows that the prime factors of 36 = 2 × 2 × 3 × 3

Factor Tree of 48

The factors of 48 can be easily listed by drawing the factor tree of 48. Let us factorize 48 using the factor tree method.

  • Step 1: We start factorizing 48 and we split it into 6 and 8.
  • Step 2: It can be seen that 6 is split into 3 and 2; whereas, 8 can be split into 4 and 2.
  • Step 3: While we circle the prime numbers till this step, we still need to factorize 4 into 2 and 2.
  • Step 4: At this step, we have reached all the prime factors of 48 which cannot be factorized any further. This completes the factor tree and the prime factors of 48 can be listed.
  • Step 5: This shows that the prime factors of 48 = 3 × 2 × 2 × 2 × 2

☛ Related Articles

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Factor Tree Examples

  1. Example 1: Fill in the missing numbers in the factor tree of 99.

    Solution: 99 can be split into 3 and 33. Then, 3 can be circled since it is a prime number and we further split 33 into 3 and 11. Therefore,

    a.) The number in the first missing blue box is 33.

    b.) The number in the second missing yellow circle is 3.

  2. Example 2: Write the factors of 16 using a factor tree.

    Solution: The factors of 16 can be calculated with the help of the factor tree.

    This shows that the prime factors of 16 = 2 × 2 × 2 × 2

  3. Example 3: State true or false.

    a.) The factors of 24 are: 2 × 2 × 2 × 3

    b.) A factor tree can be drawn by splitting the factors of a number until we reach its prime factors.

    c.) A factor tree does not find the prime factors of a number.

    Solution:

    a.) True, the factors of 24 are: 2 × 2 × 2 × 3

    b.) True, a factor tree can be drawn by splitting the factors of a number until we reach its prime factors.

    c.) False, a factor tree finds the prime factors of a number.

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Practice Questions on Factor Tree

 

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FAQs on Factor Tree

What is a Factor Tree?

A factor tree is created to find the prime factors of a number. It is a method of factorizing a number in which the factors are split and written in such a way that it looks like the branches of a tree.

How Does a Factor Tree Work?

A factor tree works according to the usual way of factorizing a number. The only distinct feature is that the factors are split and written in a way that it appears like a tree in which the branches are split. The following steps show the way in which a factor tree works. Let us understand this by factorizing the number 28.

  • Step 1: We start factorizing 28 and we split it into 2 and 14.
  • Step 2: Now, 2 is a prime number, so we will circle it and move ahead to factorize 14 further.
  • Step 3: 14 can be factorized into 2 and 7. Since 2 and 7 are prime numbers, we stop here and circle 2 and 7.
  • Step 4: At this step, we have reached all the prime factors of 28 which cannot be factorized any further. This completes the factor tree and we list all the circled prime numbers as the prime factors of 28. This is represented as 28 = 2 × 2 × 7.

What is the Factor Tree of Hundred?

The factor tree of 100 can be drawn easily using the following steps.

  • Step 1: We start factorizing 100 and we split it into 2 and 50.
  • Step 2: Now, 2 is a prime number, so we will circle it and move ahead to factorize 50.
  • Step 3: 50 can be factorized into 2 and 25.
  • Step 4: After circling 2 at this step, we will factorize 25 into 5 and 5.
  • Step 5: At this step, we reach the stage where the factors can no more be factorized. This means we can list the circled factors as the prime factors of 100 as 100 = 2 × 2 × 5 × 5.

How to Find Prime factors Using Factor Tree?

The prime factors of a number can be listed using a factor tree. We start splitting the number by listing its factors until the factors cannot be split anymore. These factors are split and written like the branches of a tree. Once we start splitting, we circle the prime numbers and move ahead to factorize the composite numbers further. After we reach the stage where the factors cannot be split any further, we come back to the circled prime factors at each step and list these numbers as the prime factors of the given number.

How to Find LCM Using Factor Tree?

The Least Common Multiple (LCM) of two numbers is the least number which is a common multiple for both the numbers. We can find the LCM of numbers using a factor tree. For example, let us find the LCM of 6 and 8 using a factor tree.

  • Step 1: We will start factorizing 6 and split it into 2 and 3.
  • Step 2: Now, since 2 and 3 are prime numbers, we will circle them as the prime factors of 6.
  • Step 3: Then, we will factorize 8 into 2 and 4. We will circle 2 since it is a prime number and we will factorize 4 further into 2 and 2.
  • Step 4: So, the prime factors of 8 are 2 × 2 × 2.
  • Step 5: Now, we have the prime factorization of 6 and 8 and this can be expressed as, 6 = 2 × 3 and 8 = 23. We will observe these factors to move further.
  • Step 6: We know that LCM is the product of the largest multiple of every prime number that is present on at least one list. For example, we have a 2 and a 3. So, we will choose the largest multiples of 2 and 3 in this list and find their product. The largest multiple of 2 here is 23 and the largest multiple of 3, in this case, is 3. This means, the LCM of 6 and 8 = 23 × 3 = 24

How to Find GCF Using Factor Tree?

The Greatest Common Factor (GCF) of two numbers can be calculated using a factor tree. For example, let us find the GCF of 20 and 30 with the help of a factor tree.

  • Step 1: We will start factorizing 20 and split it into 2 and 10.
  • Step 2: Since 2 is a prime number, we will circle it and we will factorize 10 into 2 and 5. After circling 2 and 5, we can list the prime factors of 20 as, 20 = 2 × 2 × 5.
  • Step 3: Then, we will factorize 30 into 5 and 6. Since 5 is a prime number, we will circle it and we will further factorize 6 into 2 and 3. After circling 2 and 3 we get the prime factors of 30 as, 30 = 2 × 3 × 5.
  • Step 4: Now that we have the prime factors of both the given numbers we can list them as, 20 = 2 × 2 × 5 and 30 = 2 × 3 × 5.
  • Step 5: We know that GCF is the product of the prime factors that are common to both factorizations. This means that in both factorizations, we have 2 and 5 which are the common factors and so the GCF of 20 and 30 will be the product of 2 and 5. This means, 2 × 5 = 10. Therefore, the GCF of 20 and 30 is 10.

What is the Factor Tree for 144?

The factor tree for 144 can be drawn using the following steps.

  • Step 1: Find two factors of 144. We get 2 and 72.
  • Step 2: Observe these factors to see if they are prime numbers or not. Since 2 is a prime number we will circle it and come back to it later.
  • Step 3: We will further factorize 72 into 2 and 36. We will again circle 2 at this step since 2 is a prime number and move on to factorize 36. So, 36 can be factorized as 6 and 6. Since 6 is a composite number, it can be further split into 2 and 3. We will circle 2 and 3 since they are prime numbers. This is where the factor tree ends.
  • Step 4: Now that we cannot factorize the factors any further, we will list all the circled factors which are the prime factors of 144. This means, 144 = 2 × 2 × 2 × 3 × 2 × 3.

What is the Factor Tree of 27?

In order to create a factor tree of 27, we use the following steps.

  • Step 1: First, we will factorize 27 and split it into 3 and 9.
  • Step 2: Since 3 is a prime number, we will circle it and come back to it later. Then, we will factorize 9 into 3 and 3. Since 3 is a prime number we will circle both the 3s and complete the factor tree.
  • Step 3: At this step, we reach the stage where the factors cannot be factorized any further. This means we can list the circled factors as the prime factors of 27 as 27 = 3 × 3 × 3.

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What is GCF and LCM? – Wiki Reviews

The greatest common factor (GCF) is the largest number that is a multiple of two or more numbers , and the least common multiple (LCM) is the smallest multiple of two or more numbers. …

Similarly, what is GCF in the math example? The greatest common divisor (gcd) of a set of numbers is the biggest factor that all numbers share is . For example, the numbers 12, 20, and 24 have two common divisors: 2 and 4. ... GCF is often used to find common denominators.

What is GCF for 2 and 4?

Answer: GCF 2 and 4 is 2 .

then what is the GCF x2 and x9? The common factors for the variables x2,x9 x 2 , x 9 are x⋅xx ⋅ x . The GCF for the variable part is x2 .

How to find the greatest common divisor of three numbers?

What is the first step for a GCF solution?

Example: Solving a problem with GCF

To solve this problem, the first thing we need to do is divide both numbers into prime factors of . Now, in order to calculate the GCF, we need to choose the lowest common factors, which in this case will be 2 and 3.

What is the GCF for 2 and 6?

Answer: GCF 2 and 6 is 2 .

What is GCF 2, 4 and 6? There are 2 common divisors 2, 4 and 6, i.e. 1 and 2. Therefore, the greatest common divisor of 2, 4 and 6 is 2 .

What is GCF 2 and 8? Answer: GCF 2 and 8 are 2 .

What is GCF for x2 and x6?

The common factors for x2,x6 x 2 , x 6 are x⋅xx ⋅ x . The GCF for the variable part is x2 .

What is GCF 20, 24 and 40? The common divisors for 20,24,40, 20, 24 40 , XNUMX , XNUMX are: 1,2,4 1, 2, 4 .

What is GCF for 24 and 54?

Answer: GCF 24 and 54 is 6 .

What is GCF 60 and 90? Answer: GCF 60 and 90 are 30 .

What is GCF 4?

GCF 4 and 8 Examples

Therefore, the greatest common divisor of 4 and 8 is 4. Example 2. Find the largest number that exactly divides 4 and 8. Solution: The largest number that exactly divides 4 and 8 is their greatest common factor, i.e. gcd 4 and 8. … Divisors 4 = 1, 2, 4.

How to find the GCF of two numbers using a factor tree?

How did you get the GCF and LCM of the two numbers? First find all the prime factors. For ZKF find all factors that are the same and multiply . For LCM, find all factors that are different and multiply.

What are GCF and LCM in mathematics? – Wiki Reviews

The greatest common factor (GCF) is the largest number that is a multiple of two or more numbers , and the least common multiple (LCM) is the smallest multiple of two or more numbers. …

So are GCF and HCF the same? Highest Common Factor, GCF for short, is also known as Highest Common Factor (HCF). So, yes, GCF and HCF are the same .

Also, how are GCF and LCM similar? GCF and LCM are actually quite different from and are often easily confused. When we hear these names, we tend to hear the GREATEST and LOWEST. … The least common factor will always be 1. The greatest common factor of several numbers is the highest factor (i.e., the smallest value) that is divisible by all given numbers.

How is LMB calculated? One way to find the least common multiple of two numbers is to first list the prime factors of each number . Then multiply each factor by the maximum number of times it occurs in any number. If the same factor occurs more than once in both numbers, you multiply the factor by the largest number of times it occurs.

How to find the GCF of two numbers?

To find the common divisors of two numbers, first you need to list all the factors of each, and then compare them . If a factor appears in both lists, then it is the common factor.

What is the greatest common divisor of 42 and 126? Answer: The greatest common divisor of 42, 126 and 210 is 42 . Let's solve it step by step. GCF is the greatest common divisor, or the largest integer that all numbers are divisible by. Factors of the number 42: 1, 2, 3, 6, 7, 14, 21, 42.

How to find the greatest common divisor of three numbers?

How do you use GCF in real life?

We use the largest common factors all the time with fractions , and because fractions are used a lot in everyday life, this makes GCF very useful! By finding the GCF of the denominator and numerator, you can successfully simplify a fraction or ratio. For example, we can simplify 3045 knowing that its HCF value is 15.

How to make premium?

How to find the greatest common divisor of two numbers?

To find the GCF of a set of numbers, list all factors of each number . The most important factor appearing in each list is the GCF. For example, to find the GCF of the numbers 6 and 15, first list all the divisors of each number. Since 3 is the largest factor that appears on both lists, 3 is GCF 6 and 15.

What is GCF 45? GCF 45 and 90 is equal to 45 . To calculate the greatest common factor of 45 and 90, we need to factor each number (factors 45 = 1, 3, 5, 9, 15, 45; factors 90 = 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90) and choose the largest factor that exactly divides both 45 and 90, i. e. 45.

Which GCF at 10 and 45? Answer: GCF 10 and 45 are 5 .

What is the GCF of 12 and 32?

As you can see when you list the factors of each number, 4 is the largest number that 12 and 32 are divisible by.

What is GCF 20 and 12? Answer: GCF 12 and 20 are 4 .

What is GCF 42?

GCF 42 and 54 is 6 . To calculate the greatest common factor of the numbers 42 and 54, we need to factor each number (factors 42 = 1, 2, 3, 6, 7, 14, 21, 42; factors 54 = 1, 2, 3, 6, 9, 18, 27, 54) and choose the largest factor that exactly divides both 42 and 54, i.e. 6.

What is GCF 8 and 42? As you can see when you list the factors of each number, 2 is the largest number that 8 and 42 are divisible by.

What are the 4 steps of solving word problems involving GCF and LCM?

We use a four-stage plan when solving problems related to the GCF and LCM of two given numbers. Understand, plan, decide, check and look back .


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