What is GCF?
The GCF stands for greatest common factor, it is also known as highest common factor(HCF). GCF is the greatest among the common factors of a number. The factors that are shared by two or more numbers called common factors.
Methods to Find GCF(HCF)
- Listing of Factors
In this method, we list the factors of two or more numbers and find the common factors from the list. Then we select the greatest among all the common factors.
- Prime Factorization
In this method, we use a factor tree to find the prime factorization of numbers and list out the common prime factors of the numbers. Then the product of all the common prime factors results in the greatest common factor.
What is LCM?
The LCM stands for least common multiple. The LCM is the least multiple among all the common multiples of two or more numbers.
As we already know, a multiple is the number obtained by multiplying a number by itself or any other number.
Methods to Find LCM
- Listing Multiples
In this method we list the multiples of two or more numbers and find out the common multiples from the list. Then, the least among all common multiples is the least common multiple.
- Prime Factorization
In this method we use the factor tree to find the prime factorization of the numbers. List each different factor where it appears the greatest number of times and multiply them to get the least common multiple.
Solved Examples
Example 1. Find the greatest common factor of 24 and 30 using listing the factors method.
Solution:
The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.
The factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30.
Circle the common factors.
The common factors of 24 and 30 are 1, 2, 3, and 6. Hence, the greatest common factor of 24 and 30 is 6.
Example 2. Find the greatest common factor of 28 and 44 using prime factorization.
Solution:
Use factor tree method to get prime factors of numbers:
28 = 2 x 2 x 7
44 = 2 x 2 x 11
So, the GCF of 28 and 44 is 2 x 2 = 4.
Example 3: Find the least common multiple of 4 and 5. Use the listing of multiples method.
Solution:
Multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, …
Multiples of 5 are 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, …
The common multiples have been encircled below:
The common multiples of 4 and 5 are 20, 40, and so on. The least of these common multiples is 20.
Hence the LCM of 4 and 5 is 20.
Example 4: Sam writes a poem every 2 days and Annie writes a poem every 3 days. Today both Annie and Sam are writing poems. After how many days do they write a poem together again?
Solution:
You are given the number of days Sam and Annie take to write a poem. The LCM of the number of days will give us the time when they will write together again from this day. So, let’s find the LCM of 2 and 3.
Multiples of 2 are 2, 4, 6, 8, 10, 12, 14, 16, 18, 20.
Multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24, 27, 30.
The common multiples of 2 and 3 are 6,12 and 18. The least of these common factor is 6
so, the LCM of 2 and 3 is 6.
Hence, Sam and Annie will write together again on the 6th day from the current day.
GCF or the greatest common factor is the greatest of all common factors between any two or more numbers.
LCM or the least common multiple is the least among the common multiple of two or more numbers.
GCF is the greatest number that is a common factor of two or more numbers. However, LCM is the least among the common multiple of two or more numbers.