In my various other lesson, I questioned the procedure on just how to uncover the Greatest common Factor making use of the list Method. This technique is only efficient when handling smaller numbers.
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That’s why we require to find out a backup method to identify the GCF when larger numbers are involved. This alternate an approach takes benefit of the usefulness of Prime Factorization.
I need to caution you that there is a prerequisite for this lesson. You will require to have an understanding of just how to carry out prime administrate on an integer.
Please don’t feel poor if you should take the refresher lesson. To trust me, it is easy! you will shortly realize that you are ago here in no time. Here’s the link: ☞ Integer element Factorization ☜
Steps on just how to recognize the GCF utilizing Prime Factorization
These room the actions on exactly how to uncover the greatest usual factor of 2 numbers making use of Prime Factorization. Back this method can be extended to find the GCF of many numbers, I simply want to emphasis on 2 numbers.
1) compose the element Factorization of each number. In various other words, express each number as a product of numbers composed in an exponential form. Obviously, the base will constantly be a element number.
2) identify the numbers that have actually the very same base. Ignore the exponent for now.
3) compare the exponents of the numbers through a typical base. Select the number which has actually the least exponent value. Because that instance, in 2^2 and 2^4, pick 2^2 since its exponent has a lesser worth than the of 2^4, that is, 2 .
4) multiply the numbers that you selected in action #3 to determine the greatest common factor.
Examples of identify the Greatest usual Factor (GCF) making use of Prime factorization
Example 1: What is the GCF of 36 and also 120 ?
Begin by expressing every number into its element factorized form.
Next, we determine the exponential number that have actually the exact same base. Notice that 2^2 and 2^3 have a typical base of 2. Top top the various other hand, 3^2 and 3 have actually a usual base the 3. We ignore 5 because that a basic reason that 36 doesn’t have a prime variable of 5. Again, it must be usual to both.
I require you come slow down here prior to continuing. The creature 3 appears to have no exponent. If ever you watch a optimistic integer i beg your pardon doesn’t have actually an exponent ~ above its upper right corner, don’t jump right into conclusion the it has actually an exponent the zero. That’s wrong! This is a typical pitfall of numerous of my students. The truth is, it is presume to have an exponent that 1. Therefore, 3 is same to 3^1.
For emphasis, us encircle the numbers that have a usual base making use of the very same color. Red circles for 2^2 and also 2^3 if blue for 3^2 and also 3.
Now we room going to choose which number has the the very least exponent worth for each common prime factor.
In the situation of 2^2 and 2^3, we choose 2^2 because 2 .
because that 3^2 and 3, we choose 3 because 1 .
The arrows indicated the numbers that we chose.
The final step is to multiply the numbers that we have selected indigenous the previous step. The liked numbers are those that have a common prime number base with the the very least exponent value. Therefore, the GCF that 36 and 120 is 2^2 times 3 which equals to 12.
Example 2: What is the GCF the 150 and 180 ?
This difficulty is not different from the first one. The numbers space just reasonably larger. Detect the GCF of these two numbers need to take extra work – especially with the element factorization part. However the measures remain the very same which should provide us a boost of confidence.
Start by element factorizing 150 and 180.
Now we recognize the numbers with a typical base. Lock are shade coded for emphasis.Common base of 2 and 2^2 is 2.Common basic of 3 and also 3^2 is 3.Common base of 5^2 and also 5 is 5.
For the numbers that have actually a common base, pick the number v the smallest exponent value. In between 2 and also 2^2, we made decision 2. Between 3 and also 3^2, we made decision 3. And also finally, we determined 5 in between 5^2 and 5.
For the critical step, us multiply with each other the numbers we have chosen native the previous step to recognize the GCF. Thus, the GCF the 150 and 180 is 2 time 3 time 5 i beg your pardon is same to 30.
Example 3: What is the GCF that 1,260 and 1,960 ?
Let’s take this skill of recognize the greatest common factor come the next level. This time we will discover the GCF of two numbers that have numerical values in between 1,000 and also 2,000.
By now, you’ve probably realized that finding the GCF of this numbers making use of the list technique is walk to it is in cumbersome. This is the reason why I have actually to create a different lesson on recognize the GCF utilizing the prime Factorization method. That is an ext efficient, accurate, and also prone to much less errors especially when you are working it the end by hand.
To obtain to the love of this method, you will need to have a an excellent grasp on just how to element factorize a confident integer making use of the Prime element Tree.
I can’t overemphasize the importance of Prime factor Tree. Trust me, it will certainly be your ideal friend from right here on out during your examine of algebra in general.
In a nutshell, here’s the an approach of element factorization making use of the Prime variable Tree. Start dividing the provided number through the smallest prime number i beg your pardon is 2. If 2 same divides the number, draw a diagonal line down towards the left (branch the the tree) the the given number and write 2. Then, compose the quotient by illustration a diagonal down in the direction of the right. The quotient will certainly become part of the tribe of the tree.
Keep separating the succeeding quotients by 2 while recording your outcomes as branches (divisors) and part of the trunk (quotients that are composite numbers). Store going until such time once 2 can no much longer divide the quotient. That’s as soon as you relocate to the following prime number i beg your pardon is 3, and also so on. Repeat the procedure until the quotient is a element number. This is when you stop.
❖ Here’s the Prime factor Tree the the number 1,260 and also its element Factorization.
❖ listed below is the Prime variable Tree that the number 1,960 and also its prime Factorization.
Finally, since we have properly prime factorized the two big numbers, we can now proceed as usual just like in examples #1 and also #2.Write the prime factorizations the 1260 and 1960 side by side. It’s a an excellent practice come align the numbers with the exact same base.
Identify the numbers created in exponential form that have actually the very same base. As with before, don’t mind the exponents yet.
Compare the index number of the exponential numbers having a usual base. Select the exponential number that has actually the the very least exponent value. In between 2^2 and 2^3, we determined 2^2, no 2^3, because the exponent that the former is less than the latter. Ignore 3^2 because it has actually no equivalent number with a base of 3. Because that 5 and also 5 , simply select 5 due to the fact that they are precisely the very same – a duplicate for that matter. And finally, we made decision 7 end 7^2 because that the same factor that 1 .
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The last action to gain the greatest usual factor the 1,260 and 1,960 is to main point the number we have actually selected native the vault step. Therefore, the GCF the 1,260 and 1,960 is equal to 2^2 time 5 time 7 which offers us 140.
You might also be interested in:
Finding GCF utilizing the list Method
Finding LCM using the list Method
Use element Factorization to uncover LCM
MATH SUBJECTSIntroductory AlgebraIntermediate AlgebraAdvanced AlgebraAlgebra native ProblemsGeometryIntro come Number TheoryBasic mathematics Proofs
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