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Hello,
Greetings for the day!
In our post yesterday, we discussed a simple way of understanding terms like factors and multiples. In case you haven’t seen this post, here’s a link to view the post:
https://gmatclub.com/forum/factors-multiples-a-simple-way-of-understanding-them-296311.html
While it’s true that you will get direct questions on factors and multiples, the most important application of these concepts is in HCF(GCD) and LCM.
HCF is an acronym for Highest Common Factor. It is also referred to by the name GCD, which stands for Greatest Common Divisor. The acronym LCM stands for Least/Lowest Common Multiple.
Let us start off by understanding both these in simple terms, post which, we can maybe look at the different ways of calculating these for a given set of numbers.
Highest Common Factor (HCF) / Greatest Common Divisor (GCD)
Since the word ‘Common’ is used, it is obvious that we will be dealing with at least 2 numbers. You cannot find out the HCF of a single number.
The first thing we need to understand about the HCF is that, it is a factor. This means, it’s a number that CAN divide some other numbers completely.
Followed by this, when we look at the term ‘Common Factor’, we are looking at a factor that can divide two or more numbers, commonly.
Finally, when we look at the term, ‘Highest Common Factor’, we can understand that this number is the highest of all the common factors that we found out in the previous stage.
Let’s take a simple example. Let us consider the numbers, 16 and 24.
The factors of 16 are – 1, 2, 4, 8 and 16. The factors of 24 are – 1, 2, 3, 4, 6, 8, 12 and 24.
The common factors are 1, 2, 4 and 8.
Therefore, the highest of the common factors is 8. This is the number we call as the HCF / GCD.
A simple trick to pick the HCF is to pick the last number in the list of common factors, listed in ascending order. Also, remember that since HCF is a factor, it has to be smaller than (or equal to) the numbers it has to divide. So, we can conclude that the HCF is the largest of all the common factors of a given set of numbers.
Lowest/Least Common Multiple (LCM)
On similar lines, the first thing we need to understand about the LCM is that it is a multiple. This means that, it can be found in the multiplication tables of certain numbers.
Next, the term ‘Common multiple’ means this is a number which can be found in the multiplication tables of two or more numbers, commonly.
Therefore, ‘Lowest/Least Common Multiple’ is the smallest of the common multiples which were found in the previous step.
Let us take the same numbers, 16 and 24.
The multiples of 16 are – 16, 32, 48, 64, 80, 96,….. and so on.
The multiples of 24 are – 24, 48, 72, 96, 120,…. and so on.
The common multiples are – 48, 96, 144,… and so on.
The lowest of these common multiples, is 48. This is the Least Common Multiple or LCM.
A simple trick to pick the LCM is to pick the first number in the list of common multiples, listed in ascending order. Since the LCM is a multiple, it has to be bigger (or equal) than the numbers by which it can be divided. Therefore, the LCM is the smallest of the common multiples of a given set of numbers.
Well, we hope that this post has simplified the way in which HCF and LCM can be understood. What next?
Obviously, the method that we adopted in the examples above is lengthy and cumbersome. Also, it was only meant to simplify the process of understanding HCF and LCM.
In the third part of this post tomorrow, we will discuss a few commonly used methods to calculate HCF and LCM. We will also be covering some very important concepts on HCF and LCM, which are tested frequently in the Quant section.
Thank you!
Greetings for the day!
In our post yesterday, we discussed a simple way of understanding terms like factors and multiples. In case you haven’t seen this post, here’s a link to view the post:
https://gmatclub.com/forum/factors-multiples-a-simple-way-of-understanding-them-296311.html
While it’s true that you will get direct questions on factors and multiples, the most important application of these concepts is in HCF(GCD) and LCM.
HCF is an acronym for Highest Common Factor. It is also referred to by the name GCD, which stands for Greatest Common Divisor. The acronym LCM stands for Least/Lowest Common Multiple.
Let us start off by understanding both these in simple terms, post which, we can maybe look at the different ways of calculating these for a given set of numbers.
Highest Common Factor (HCF) / Greatest Common Divisor (GCD)
Since the word ‘Common’ is used, it is obvious that we will be dealing with at least 2 numbers. You cannot find out the HCF of a single number.
The first thing we need to understand about the HCF is that, it is a factor. This means, it’s a number that CAN divide some other numbers completely.
Followed by this, when we look at the term ‘Common Factor’, we are looking at a factor that can divide two or more numbers, commonly.
Finally, when we look at the term, ‘Highest Common Factor’, we can understand that this number is the highest of all the common factors that we found out in the previous stage.
Let’s take a simple example. Let us consider the numbers, 16 and 24.
The factors of 16 are – 1, 2, 4, 8 and 16. The factors of 24 are – 1, 2, 3, 4, 6, 8, 12 and 24.
The common factors are 1, 2, 4 and 8.
Therefore, the highest of the common factors is 8. This is the number we call as the HCF / GCD.
A simple trick to pick the HCF is to pick the last number in the list of common factors, listed in ascending order. Also, remember that since HCF is a factor, it has to be smaller than (or equal to) the numbers it has to divide. So, we can conclude that the HCF is the largest of all the common factors of a given set of numbers.
Lowest/Least Common Multiple (LCM)
On similar lines, the first thing we need to understand about the LCM is that it is a multiple. This means that, it can be found in the multiplication tables of certain numbers.
Next, the term ‘Common multiple’ means this is a number which can be found in the multiplication tables of two or more numbers, commonly.
Therefore, ‘Lowest/Least Common Multiple’ is the smallest of the common multiples which were found in the previous step.
Let us take the same numbers, 16 and 24.
The multiples of 16 are – 16, 32, 48, 64, 80, 96,….. and so on.
The multiples of 24 are – 24, 48, 72, 96, 120,…. and so on.
The common multiples are – 48, 96, 144,… and so on.
The lowest of these common multiples, is 48. This is the Least Common Multiple or LCM.
A simple trick to pick the LCM is to pick the first number in the list of common multiples, listed in ascending order. Since the LCM is a multiple, it has to be bigger (or equal) than the numbers by which it can be divided. Therefore, the LCM is the smallest of the common multiples of a given set of numbers.
Well, we hope that this post has simplified the way in which HCF and LCM can be understood. What next?
Obviously, the method that we adopted in the examples above is lengthy and cumbersome. Also, it was only meant to simplify the process of understanding HCF and LCM.
In the third part of this post tomorrow, we will discuss a few commonly used methods to calculate HCF and LCM. We will also be covering some very important concepts on HCF and LCM, which are tested frequently in the Quant section.
Thank you!
