The GCD of two or more nonzero integers, x and y, is the largest positive integer m that divides x and y. Ogreatest common divisorit is commonly known as GCF. Here the largest can be replaced by the largest and the factor can be replaced by the divisor. Therefore, the greatest common factor is also known as the greatest common factor (HCD), greatest common factor (HCF) or greatest common factor (GCD).
The GCF is almost always used with fractions, which are used a lot in everyday life. To truncate a fraction or ratio, you can find the GCF of the denominator and numerator to get the truncated form you need. Also, when we look around, the arrangement of something into rows and columns, distribution and grouping, all of that requires an understanding of the GCF.
|1.||What is the greatest common factor (GCF)?|
|2.||How do I find the GCF?|
|3.||GCF e LCM|
|4.||Frequently asked questions about the GCF|
What is the greatest common factor (GCF)?
OGCF (greatest common factor)of two or more numbers is the greatest number common to allfactorsthe given numbers. The gcd of two natural numbers x and y is the largest possible number that divides x and y without leaving anythingRest. To calculate the GCF, there are three common methods: division,multiplication, and prime factorization.
Example:Let's find the greatest common factor of 18 and 27.
First, we list the factors of 18 and 27, and then we figure out the common factors.
Factor 18: 1, 2, 3, 6, 9, 18
Factor 27: 1, 3, 9, 27
The common divisors of 18 and 27 are 1, 3 and 9. Among these numbers, 9 is the largest (biggest) number. And that's what happenedGCF the 18th and 27this 9. This is written as: GCF(18, 27) = 9.
A divisor of a number is youDivisorlike this. Therefore, the greatest common divisor is also calledgreatest common divisor(you) GCD.In the above example, the greatest common divisor (gcd) of 18 and 27 is 9, which can be written as:
ggT (18, 27) = 9.
How do I find the GCF?
Following are the three methods for finding the greatest common divisor of two numbers:
- list common factors
- prime factorization
- division method
GCF listing factors
In this method, the factors of both numbers can be listed, so it's easy to look for common factors. By scoring the common factors, we can choose the greatest among them all. Let's look at the example given below:
Example:What is the GCF of 30 and 42?
- Step 1 - List the factors of each number.
- Step 2 - Check all the common factors.
- Step 3 - 6 is the common and greatest factor.
For this reason,GCF the 30th and 42nd= 6. This method can also be used to find GCF of three or more numbers.
Finding the greatest common factor by listing factors can be difficult when the numbers are larger. In these cases we use theprime factorizationand division methods to find GCF.
GCF by prime factorization
Prime factorization is a way of expressing a number as the product of its prime factors, starting with the smallest prime factor of that number. Let's look at the example given below:
Example:What is the GCF of 60 and 90?
- Step 1 - Represent the numbers in prime factor form.
- Step 2 - GCD is the product of the factors common to each of the given numbers.
So GCF(60,90) = 21× 31× 51= 30. Therefore,GCF the 60s and 90s= 30. We can also use this method to find the greatest common divisor of three or more numbers.
Finding GCF by the division method
Division is a method of grouping objects into equal groups as we proceed to large numberslong division, which breaks a division problem into a series of simpler steps. The greatest common factor (GCF) of a set ofwhole numbersIt's the biggestpositive integerwhich divides all given numbers without leaving a remainder. Let's look at the example given below:
Example:Find the GCF of 198 and 360 using the division method.
Between the two given numbers, 360 is the larger number and 198 is the smaller number.
- Step 1 - Divide the largest number by the smallest long division number.
- Step 2 - If the remainder is 0, then the divisor is the GCF. If the remainder is not 0, make the remainder from the step above the divisor and the divisor from the step above toDividingand perform long division again.
- Step 3 - If the remainder is 0, then the divisor of the last division is the GCF. If the remainder is not 0, we have to repeat step 2 until we get the remainder 0.
Therefore, the GCD of the two given numbers is the divisor of the last division. In this case, the divisor of the last division is 18. Therefore, the GCF of 198 and 360 is 18. This method is the most convenient way to find the GCF of large numbers. Let's see how to use the division method to find the greatest common divisor of three numbers. To find the GCD of three numbers by long division, the following steps must be followed:
- First, we find the GCD of two of the numbers.
- Then we find the GCF of the third number and the GCF of the first two numbers.
Example:Find the GCF of 126, 162 and 180.
First, we find the GCD of the two numbers 126 and 162. [You can choose any two numbers from the three numbers provided]
So GCF of 126 and 162 = 18........(1).
Then we find the GCF of the third number which is 180 and the above GCF 18 using the same method.
So GCF of 180 and 18 = 18......(2).
From (1) and (2) GCF(126, 162, 180) = 18. So GCF from 126, 162 and 180 = 18.
GCF e LCM
The greatest common divisor is the greatest number that divides the given numbers without leaving a remainder. On the other hand, theLCM (Least Common Multiple)is the least commonseveralof the given numbers that can be exactly divided by the given numbers without leaving a remainder. For example, let's find the GCF and LCM of numbers 6 and 8.
Ofactor 6are 1, 2, 3, 6 and thefactor 8are 1, 2, 4, 8. So the common divisors of 6 and 8 are 1 and 2, of which 2 is the greatest common divisor. So GCF(6, 8) = 2. Now the firstmultiples of 6are 6, 12, 18, 24, 30, ..., and the firstmultiples of 8are 8, 16, 24, 32, ... Of these, the smallestcommon multipleof 6 and 8 is 24. So LCM(6, 8) = 24.
a very interestingRelationship between GCF and LCMof two numbers is that the product of GCF and LCM of two numbers is equal toproductsthe numbers. For any two numbers a and b, LCM (a, b) × GCF (a, b) = a × b. Let's verify this using the example from 6 and 8 above. Let a = 6 and b = 8.
LCM (6, 8) × GCF (6, 8) = 6 × 8
24 × 2 = 6 × 8
48 = 48
Now let's know the difference between GCF and LCM in the following section.
Difference between GCF and LCM
The GCF or greatest common divisor of two or more numbers is the greatest divisor among all the common divisors of the given numbers, while the LCM or least common multiple of two or more numbers is the smallest number among all the common multiples of the given numbers. The table below shows the difference between GCF and LCM:
|Greatest Common Factor (GCF)||Least Common Multiple (LCM)|
The gcd of two natural numbers a and b is the largest natural number x that is a factor of a and b.
O LCM of two natural numbers a and b is the smallest number y that is a multiple of a and b.
At the intersection of sets of common factors, it is the largest value.
At the intersection of sets of common multiples, it is the minimum value.
|Represented as GCF(a,b) = x|
Represented as LCM(a,b) = y
► Related Topics
Check out these articles on the greatest common factor (GCF) concept in math.
- Factoring Methods
- GCF calculator
Frequently asked questions about the GCF
What is the greatest common factor (GCF)?
The greatest number of allusual factorsof two or more numbers is called the greatest common divisor or GCF. For twoPay, the GCF is the greater number dividing the two given numbers. The GCF can be calculated using the basearithmetic operationsin math, ie H. Division, multiplication and prime factoring.
How to find the greatest common factor (GCF)?
Find the greatest common divisor of two or morenatural numbers, there are 3 methods that can be used - listing the common factors, primary factoring anddivisionMethod. Each method requires division and multiplication to obtain the GCF. for example theGCF the 14th 35this 7. Using the common factor listing method, theFactor 14are 1, 2, 7, 14 and theFactor 35are 1, 5, 7, 35. The two common factors are 1 and 7, of which 7 is the greater. So 7 is the GCF of 14 and 35.
What is the greatest common divisor of two prime numbers?
A prime number has only two divisors (1 and itself). then twoPrime numberscannot have a common divisor other than 1. Therefore, the greatest common divisor of two prime numbers is always 1. For example, thegreatest common divisor of 5 and 7is 1
What is the greatest common factor of 24 and 54?
OFactor 24are 1, 2, 3, 4, 6, 8, 12 and 24.Factor 54are 1, 2, 3, 6, 9, 18, 27, and 54. The common divisors of 24 and 54 are 1, 2, 3, and 6.GCF the 24th 54thit's 6
What is the GCF of 15 and 20?
OFactor 15are 1, 3, 5 and 15. Thefactor 20are 1, 2, 4, 5, 10 and 20. The common divisors of 15 and 20 are 1 and 5.GCF the 15th and 20this 5
How do I find GCF and LCM?
GCF (greatest common factor) and LCM (least common multiple) can be found using one of the methods described below:
- listing method
- prime factorization method
- division method
All methods are used differently for GCF and LCM.
Are GCF and HCF the same?
The greatest common factor is abbreviated as GCF and is also known asgreatest common divisor(HCF). So yes, GCF and HCF are the same thing.
Is GCF greater than LCM?
The LCM is the least common multiple of the given numbers that can be divided by both numbers, while the GCF is the greatest common divisor of the given numbers that both numbers share. Thus, for any two numbers, the LCM of the numbers is greater than the GCF of the numbers.
How do you find the greatest common divisor of a polynomial?
The greatest common divisor of apolynomialcan be found by following the steps below:
- Step 1:Carefully observe all the terms of the given polynomial.
- Step 2:Find the numbers or variables that are common to all terms.
This will be the required GCF of the polynomial.
How do you factor the GCF?
The GCF of two or more numbers can be obtained using the primary factorization method, which only takes a few steps. They are:
- List the prime factors of all numbers.
- Circle the common prime factors of all the numbers.
- Multiply all the circled numbers together to find the GCF.