Introduction
GCF and LCM are terms used in mathematics. They are related to factors and multiples. The full form of GCF is the Greatest Common Factor, and the full form of LCM is the Least Common Multiple. The GCF is also called the HCF, whose full form is Highest Common Factor. A number consists of many factors. A factor is a number that, when multiplied with another number, which is also a factor of that number, gives the concerned number as the multiple. 12, e.g., has the factors 1, 2, 3, 4, 6, and lastly 12 itself. But 12 is also a multiple of all these factors as multiple is the product of the multiplication of two factors, such as 2*6 is 12.
Factors, on the other hand, are whole numbers. So, the greatest common factor among all the common factors of a number is the GCF of that number, and factors that are common between two or more numbers are called common factors. Coming to multiples, there can be multiples like factors that are common between two or more numbers, and from that list of common multiples, the least common multiple is the LCM of the concerned numbers.
GCF vs LCM
The full form of GCF is the Greatest Common Factor. It is also called HCF, which means Highest Common Factor. LCM stands for Least Common Multiple. The main difference between GCF and LCM is that GCF is concerned with factors, whereas LCM is concerned with multiples. A number consists of many factors, and when a factor is common between two or more numbers, it is called a common factor. So, among all the common factors of the concerned numbers, the greatest is called the GCF or the HCF. Like factors, a number also consists of many multiples and the multiples common between two or more numbers are called common multiples of the concerned numbers. The least common multiple in the list of common multiples becomes the LCM of the numbers. Both GCF and LCM are found through the prime factorization method.
Difference Between GCF and LCM in Tabular Form
| Parameters of Comparison | GCF | LCM |
| Full Form | The full form of GCF is the Greatest Common Factor. | The full form of LCM is the Least or the Lowest Common Multiple. |
| Meaning | The greatest or the highest common factor in the list of all the common factors of the concerned numbers is the GCF or HCF of the numbers | The lowest common multiple in the list of all the common multiples is the LCM of the concerned numbers. |
| Other Names | GCF is also called the Greatest Common Divider and Highest Common Factor or HCF. | No other names are used for LCM. |
| Type of Relation with the Numbers | The GCF must be equal or smaller than the numbers that are being compared. | LCM is always equal or greater than the numbers being compared because it is a multiple, product of multiplication of the two numbers which are also its factors. |
| Places Where It is Used | GCF is used to solve equations, for simplifying fractions, and also in factorization. | LCM is used in problems regarding multiples such as when fractions with different denominators are to be added or when a common time is required for events. |
| Prime Factorization Method | In the prime factorization method, to find the GCF of the required numbers, the numbers need to be broken down separately into their prime factors and from this list of prime factors find the common prime factors. The common prime factors with the lowest exponent should be taken from each number and then the multiplication product of these prime factors is the GCF. | To find the LCM, the prime factorization method is also used. The numbers will be broken down into their prime factors. Each prime factor's highest power should be taken and then they will be multiplied to obtain the LCM. |
What is GCF?
The full form of GCF is the Greatest Common Factor. It is the greatest or the highest common factor among all the other common factors of two or more numbers. A number consists of many factors. A factor is a whole number which, when multiplied with another factor, gives the multiple. E.g., the factors of 12 are 1, 2, 3, 4, 6, and 12. These are factors of 12 because their product will give the result of 12. But 12 is a multiple of all these numbers, which are factors of 12. A multiple is a product of the multiplication of two factors. So, when a factor is common between two or more numbers, it becomes a common factor of the concerned numbers. So, GCF is the largest common factor among all the common factors. The GCF or the HCF is a factor of the numbers which are multiples, so, it divides the multiples into two or more equal halves.
This is because the GCF is equal to or smaller than the concerned numbers, so it can divide the multiples, and it is known that multiples are divisible by factors. GCF is used for solving equations and simplifying fractions to make the process of calculation easier. The GCF should be a prime number. A prime number is a number that can be divided by 1 and by itself. GCF is also known as HCF and Greatest Common Divider. HCF stands for Highest Common Factor.
Types of Methods of Finding GCF
There are three types of methods of finding the GCF which are: -
- Prime Factorization Method
- Division Method
- Factorization Method
Prime Factorization Method
In the process of finding out the GCF through the prime factorization method, the numbers are broken down into their prime factors. Then, from this list of prime factors, the prime factors common between all the numbers are selected. Finally, the common prime factors with the lowest exponent are taken from each number, and their multiplication gives the GCF
Division Method
Suppose the GCF of 12 and 20 is to be found through the division method. The smallest number here is taken as the divisor, which is 12 here, and the bigger number becomes the dividend, which is 20 here, like the usual division method. The dividend is divided by the divisor. So, 20 is divided by 12, which brings the remainder to 8. The divisor is further divided by the remainder here, and this goes on until the remainder is 0. So, 12 divided by 8 gives 4 as the remainder. 8 is then divided by 4, which brings the remainder to 0. Hence, the GCF of 12 and 20 is 4.
Factorization Method
In the factorization method, the factors of the different numbers are listed down. Then, from among the list of factors, the highest factor, which is common among all the numbers, is the GCF of the numbers.
What is LCM?
The full form of LCM is the Least Common Multiple or Lowest Common Multiple. In LCM, the lowest common multiple is found from a list of multiples that are common between two or more numbers. A multiple is the product of the multiplication of two factors. E.g., 12 is a multiple of its factors which are 1, 2, 3, 4, 6, and 12. The LCM can be divided by concerned numbers because the LCM has to be equal to or larger than the numbers as it is a multiple. LCM is used to solve problems regarding multiples, such as when fractions with different denominators are to be added or when a common time is required for events.
Types of Methods of Finding LCM
The different types of methods used to find LCM are
- Common Multiple Method
- Prime Factor Method
- Common Division Method
Common Multiple Method
The most useful mathematical method to find the LCM of two or more given numbers is considered to be the Common Multiple Method. Here, the multiple that is the lowest or the least among all the given multiples of the numbers is taken as the LCM of the numbers. Supposedly, the LCM of 2, 3, and 4 is to be found. Firstly, some of the multiples of all the three numbers are to be written, and from this list of multiples of all the three numbers, the common multiples are to be identified. But LCM will be the multiple that is the smallest or the lowest among all the other common multiples. So, comparing the multiples of all three numbers brings 12 as the smallest common multiple among all the other common multiples. Hence, 12 is the LCM of 2,3 and 4.
Prime Factor Method
To find the LCM, the prime factorization method is also used. The numbers will be broken down into their prime factors. Each prime factor's highest power should be taken, and then they will be multiplied to obtain the LCM. Suppose the LCM of 8, 12, and 24 is to be found out. Firstly, the numbers will be broken down into their prime factors through the prime factorization method. The prime factors will be written in exponential notation, and then from the three numbers, the prime factors having the highest exponential notation should be selected. Their multiplication will give the LCM which is 12 in this case.
Common Division Method
The simplest method of finding the LCM is the common division method, where the numbers are divided by a single number to find their LCM. Firstly, perpendicularly intersected lines are drawn. Then, above the horizontal line on the right-hand side, write the numbers, and on the left-hand side, the number with which all of them will be divided. The number with which all of them will be divided should be able to divide at least two numbers. The remainder will also be divided until there are numbers in which at least two numbers cannot be divided anymore. Then, the product of the divisors is to be multiplied with the undivided numbers, which gives the LCM of the numbers.
Major Differences Between GCF and LCM (In Points)
- The full form of GCF is the Greatest Common Factor, and the full form of LCM is the Least or Lowest Common Multiple. The basic difference between GCF and LCM is that GCF is concerned with factors, and LCM is concerned with multiples. A number consists of many factors. Factors are whole numbers, and the product of two factors is a multiple. When a factor is common between two or more numbers, it is called a common factor. The greatest or the highest among the list of common factors is the GCF, whereas in a set of common multiples of two or more numbers, the lowest or the least one is the LCM of the numbers. GCF is also called the Greatest Common Divider or Highest Common Factor (HCF).
- A GCF has to be equal to or smaller than the numbers whose GCF is calculated because it is a factor of the numbers that are multiples. LCM, on the other hand, has to be equal to or greater than the numbers because it is a multiple of the numbers, which are its factors. GCF is used in factorization and also for solving equations. LCM, on the other hand, is used for problems regarding multiples, such as when fractions with different denominators are to be added or when a common time is required for the events.
Conclusion
Hence, GCF and LCM are terms used in mathematics. The full form of GCF is Greatest Common Factor, whereas the full form of LCM is Lowest Common Multiple. The primary difference between GCF and LCM is that GCF is related to factors, whereas LCM is related to multiples. A number consists of many factors as well as multiples. The greatest or the highest common factor among a list of common factors is the GCF of the numbers, and the lowest common multiple among a list of multiples is the LCM of the numbers. The GCF has to be equal to or smaller than the numbers because it is a factor, whereas the LCM has to be equal to or greater than the numbers. After all, it is a multiple.
References
- https://calcworkshop.com/fractions/gcf-lcm/
- https://www.mathdoubts.com/methods-of-finding-lcm/
- https://ccssmathanswers.com/method-of-hcf/
