Definition Of A Common Factor

Kickoff things start, let us clear some terminologies before we discuss what Highest Common Factor is.

Gene VS Divisor

To come across their differences, let's take a look at the illustrations below.

In the first prototype, we tin see that the divisor (4) tin can divide the dividend (vi), but there is a balance. Hence, a divisor is any number that can divide the dividend with the possibility of having a residual.

Nonetheless, past looking at the second image, 2 (divisor) tin can split up half dozen (dividend) exactly. And as nosotros all know, 2 is a factor of vi. Therefore, a factor is a kind of divisor that can divide the dividend exactly or without rest.

Factor VS Multiple

These terms are commonly interchanged, but they are not the same. Let us examine the tabular array of multiples of 5 below to know why.

From the table of multiples of five, we can see that factors are numbers being multiplied to get a production. While multiples are products obtained by multiplying factors. Multiples can also exist calculated when you skip count with a number.

To practise your mind with these terms, permit united states have this activity beginning.

Which of the post-obit are the divisors or 10?

two

Reply: All of them. Y'all can utilize all those numbers to divide 10. Every bit for 1 and 2, yous can divide 10 and go exact quotients, while y'all can nevertheless have a balance when you lot utilise 9 as the divisor.

Which of the following is/are the factor(s) of 10?

Answer: Both

10 = 10 x 1

10 = 5 x ii

Which of the following is/are the multiple(s) of ten?

fifteen

twenty

Answer: 10 and 20; Remember, multiples are products, not factors. Furthermore, multiples of a number should kickoff from itself, so any number that is lower from it is non considered equally its multiple.

10 = 10 ten one

20 = 10 x ii

In a nutshell, a divisor includes all numbers that tin can separate a number either with a remainder or not. Since a factor is used to multiply a certain number, it tin also divide that number evenly or without remainder. Moreover, multiples is a series of products obtained from multiplying factors, which one from each prepare of factors is the same number.

Definition of Highest Common Gene

The Highest Mutual Cistron or HCF (too known as the Greatest Common Factor or GCF) is the highest gene that two or more numbers share or have in mutual. Information technology has a notation of HCF (a, b) = n , where a and b are the numbers that have HCF and due north as their HCF. There are dissimilar methods to obtain the HCF of two or more than numbers. Yes, in many cases, yous demand to get the HCF of two or more than numbers to solve a trouble. Merely do not worry because we are going to stick with the basics starting time to gear up the foundation and discuss later the quick ways to find the HCF of certain numbers.

Methods of Obtaining the Highest Mutual Factor or HCF

Method ane: Listing Method

This method is applicable for finding the Highest Mutual Cistron of small numbers. We can practise information technology by listing the factors of each number to find the greatest amongst them.

Example 1:

HCF {4, 10) = ?

four = {i, 2, iv}

10 = {1, 2, 5, 10}

We can run across from the lists of the factors that they share the aforementioned gene of 2. Since it is the merely factor, this is the Highest Common Gene by default.

four = {ane, 2, 4}

10 = {i, 2, 5, 10}

HCF (4, 10) = 2

Example 2:

HCF (sixteen, 18) = ?

ix = {1, 3, 9}

18 = {1, ii, 3, half-dozen, 9, 18}

nine and xviii accept common factors of 3 and 9. Nevertheless, nine is higher than 3. This means that is 9 their Highest Common Factor.

nine = {1, 3, 9}

xviii = {1, ii, 3, six, nine, 18}

HCF (9, xviii) = ix

Example 3:

HCF (11, 25) = ?

11 = {1, 11}

25 = {1, 5, 25}

HCF (xi, 25) = 1

Method 2: Prime Factorization Using Factor Tree and Tabular Sectionalisation

From the word itself, Prime Factorization is a method of getting the factors of a number that are prime numbers. Whether we are using Cistron Tree or Tabular Segmentation, our goal is to end up with prime factors, multiply them and declare it as the HCF. The first five prime number numbers are 2, three, v, 7, and eleven.

Note:
1 is non a prime considering by definition, a prime number must accept 1 and the number itself (which must exist a different number from one) as factors.

But in the instance of i, the factors of information technology are 1 and one besides. Unlike the smallest prime number, ii, which has factors of ane and 2 (which is a different number from i).

For this reason, we are only going to stop splitting the factors until the final factors listed in the factor tree are prime numbers.

Example 1: Using a Cistron Tree

xv and 18

To get the prime factors of 15 and 18 let us commencement with making a gene tree for each i of them to break down their factors.

Then, we tin can interpret the cistron trees into equations by merely getting the prime factors. This ways that for 18, we must not include nine, because it is already split past 3 3.

15 = three × 5

xviii = 2 3 3

Lastly, nosotros can see that 3 is the just factor that 15 and xviii accept in common.

15 = three × five

18 = 2 three three

This means that 3 is the Highest Common Factor of 15 and 18 or HCF (fifteen, xviii) = iii.

Example 2: Using a Table

Let the states utilize 15 and 18 again to encounter the departure of this method from the Factor Tree method. For the Table Method, we demand to make a table with two columns putting the number to be factored on the meridian right.

And then, write the everyman prime number factor of 15 on the top left. Here, we have 3.

And so, carve up 15 past iii, and write the quotient below fifteen.

Echo the process by writing the lowest prime gene of 5 on the left- hand side and calculating their quotient.

Now that we had reached 1, we ran out of prime factors. This means that we can terminate now the procedure. We may at present translate it into an equation by collecting the prime factors in the first cavalcade of the tabular array.

15 = 3 × 5

Permit u.s. do the aforementioned process for 18.

Then, equate 18 to its prime factors.

18 = iii x 2 x 3

Going back, here are the prime factors of 15 and 18.

15 = 3 × five

18 = 3 ten 2 x 3

We can also notice here that since iii is their only common factor, information technology is likewise their Highest Common Cistron.

xv = three × 5

eighteen = 3 ii iii

HCF (xv, 18) = three

This fashion of Prime number Factorization is more than efficient in finding the HCF of large numbers. Let united states of america run across the proof in the next instance.

Example 3:

HCF (120, 300) = ?

Since 120 and 300 are both even numbers, we can prepare 2 as their lowest prime factors. And then, let us starting time with dividing both of them by ii, and echo the procedure until reaching the quotient of 1.

Now, we can equate each one of them to their prime number factors.

120 = 2 10 two × 2 x 3 x 5

300 = 2 x ii 10 iii x five x v

Note that their common factors are more than than one: 2, 3, and 5. For these cases, we must non declare the greatest among them as the Highest Mutual Gene. So, nosotros cannot declare 5 equally the HCF of 120 and 300. Instead, we are going to collect the identical numbers that are contrary from the equations and the remaining common factor.

120 = 2 x 2 × 2 x iii 10 5

300 = 2 x 2 x 3 x 5 10 5

HCF (120, 300) = 2 x two x five x 3

HCF (120, 300) = sixty

Method iii: Continuous Partitioning

We can also do the Tabular Method by dividing both numbers in the same table. This method is well known equally the Continuous Partitioning, a technique that is useful if the pair are multiples of or divisible by the aforementioned number.

Offset, let us have a epitomize of the first v prime numbers:

two, three, 5, seven, 11

Since those are the first five, they are too the everyman prime numbers, numbers that we can use as divisors for the pair of numbers. The similarity from the previous method is that we divide each number by their lowest prime factor, simply for Continuous Division Method we are going to divide both numbers by their common prime number gene.

Let united states of america utilise 120 and 300 again using the Continuous Division Method.

Beginning, write the three columns putting 120 and 300 to the last two cells on superlative.

And then, divide the numbers by the prime factor that is too divisible past them. 120 and 300 are both divisible by 2 (prime number), and then we are going to utilize two every bit the first divisor.

Next, write their quotients below their respective dividends. Since lx and 150 are both fifty-fifty numbers, nosotros can nevertheless divide them by ii.

Repeat the process until we ran out of numbers to exist divided by the same prime number number.

Since 2 and 5 cannot be divided by any same prime number, we may now stop the procedure.

Then, we are going to multiply all the prime factors we got in the first cavalcade.

HCF (120, 300) = two x 2 x 3 × 5

HCF (120, 300) = 60

Annotation: Nosotros tin can also apply this method as a shortcut to observe the HCF of pocket-size numbers. Allow us use xv and eighteen again every bit an example.

15 and 18 have the lowest prime cistron of iii. Then, we are going to use 3 equally the divisor.

Then, calculate their quotients and put them beneath their respective dividends.

At this signal, nosotros can now end the process considering 5 and 6 cannot be divided past the same prime factor. Also, we are left only with one prime number factor of 3. This means that 3 is the HCF of 15 and 18.

If we are given pairs of numbers that are multiples of or divisible by the same number, Continuous Partitioning is the best strategy to chop-chop find their HCF. Merely for these cases, we can proceed past using the number that all of them are multiples of or divisible by as the first divisor.

Instance:

HCF (16, 48) = ?

Since 16 and 48 are both multiples of 16, we are going to utilise 16 as the offset divisor.

Then write the quotients below their respective dividends.

We can find hither that nosotros are already left with numbers that cannot exist divided past the same number. This means that we can now end the process and set the but divisor we have, 16, every bit the Highest Mutual Factor of sixteen and 48.

Therefore, HCF (16, 48) = 16.

Method 4: Euclidean Algorithm Using Long Segmentation

We can also use Long Division to find the Highest Mutual Factor of two larger numbers and seems not multiples of the same number at a first sight. Nosotros tin can practise this past dividing the larger number by the smaller number. Side by side, use the remainder as the new divisor to split the preceding divisor. Then, repeat the process until there is no residuum. The last divisor is the Highest Mutual Cistron of the pair of numbers.

Instance:

HCF (105, 189) = ?

Since the last divisor that nosotros got is 21, we can say that the Highest Common Factor of 105 and 189 is 21 or HCF (105, 189) = 21.

Annotation: Since we talked about different strategies on how to find the Highest Mutual Gene of a pair of numbers, the all-time technique still is whatever yous are comfortable with. You may notice that even though we used the same examples using different methods, we can still obtain the same HCF.

Getting the Highest Common Gene of Three or Four Numbers

To efficiently find the Highest Common Factors of 3 numbers, we tin use both Continuous Sectionalisation Method and Euclidean Algorithm using Long Division.

Example one: HCF (27, 45, 81)

Using Continuous Partitioning

27, 45, are, 81 are both multiples of nine. So, we are going to use 9 as their divisor.

Then, divide the numbers by 9 and put the quotients below their respective dividends.

three, 5, and 27 cannot be divided by the same prime factor, so we may at present finish the process. Since nosotros only take 9 as the divisor, it means that it is also their Highest Mutual Factor.

Hence, HCF (27, 45, 81) = ix.

Using Long Sectionalisation

First, choose ii numbers to divide. Ideally, cull the numbers that are non a factor of the other. For instance, we cannot choose 27 and 81 as the first pair because 27 is a factor of 81 and nosotros might obtain incorrect HCF. So we are going to divide 81 past 45 first.

Now that in that location is no remainder left, nosotros tin now end the process for this pair. Since we have 9 as the last divisor, nosotros are now going to use this equally the divisor to split the remaining number which is 27.

Considering we notwithstanding get 9 as the last divisor, this means that ix is the HCF of the iii numbers.

Therefore, HCF (27, 45, 81) = 9.

Nosotros can notice that either way, whether we use Continuous Division or Long Segmentation, we can nonetheless become the same answer.

Example 2: HCF (24, 48, 84, 96) = ?

If we need to find the HCF of four numbers, it is efficient to use a quicker technique. Since all of the given numbers are even, nosotros can use Continuous Partitioning.

Then, multiply the prime factors to get the HCF.

HCF (24, 48, 84, 96) = 2 10 2 x 3

HCF (24, 48, 84, 96) = 12

Applications of Highest Common Factor

There are lots of real-life situations where nosotros can use finding the Highest Common Factor of numbers. But the common denominator of these cases is having a goal of distributing things evenly to a unlike ready of numbers.

Instance:

Mrs. Gomez needs to choose students from 3 sections with 12, 24, and 30 students to correspond their sections for a school activity. She does non want to depend on how large or pocket-size the population of each section is. Rather, she wants to exist off-white past choosing the same or the common number of students from each department. How many students volition she choose from each section?

Solution:

We need to discover the HCF of the three numbers, 12, 24, and 30 to find the highest mutual number that Mrs. Gomez will pull out from each department.

12, 24, and 30 are divisible by 6, so it is fine to proceed with using it equally the offset divisor.

Since the quotients cannot be divided by any same factors anymore, we tin declare half-dozen every bit the HCF. Therefore, Mrs. Gomez should choose 6 students from each department to make a fair count.

Activity

Directions: Find the HCF of the following pair or grouping of numbers. Use the method or strategy you are most comfy with or you find suitable for the case of the numbers.

one. HCF (18, 51) = ?	2. HCF (xiv, 112) = ?
iii. HCF (20, 110) = ?	4. HCF (49, 98) = ?
five. HCF (160, 300) = ?	6. HCF (138, 437) = ?
seven. HCF (13, 26, 53) = ?	8. HCF (36, 90, 729) = ?
9. HCF (75, 100, 175, 200) = ?	ten. HCF (16, 52, eighty, 124) = ?

Solutions:

Summary

A factor is a kind of divisor that divides a number exactly. Moreover, a cistron is different from multiple, considering multiples are products of factors.
The Highest Mutual Gene is the highest cistron that tin can divide a pair or group of numbers equally or without balance.
There are dissimilar ways to obtain the HCF or GCF of numbers:
- List Method
- Prime Factorization
  - Factor Tree
  - Tabular Partitioning
- Continuous Partitioning
- Euclidean Algorithm Using Long Sectionalisation
In real-life situations, you can utilise the principle of HCF on finding the mutual number or corporeality you want to obtain from a gear up of different numbers.

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