In our second part of this post, we showed you how you can understand the concepts of HCF and LCM in a simple manner. In doing so, we also consolidated our understanding of the terms factors and multiples.
In case you missed out on the previous two posts in this series, here are the links to them:
https://gmatclub.com/forum/factors-multiples-a-simple-way-of-understanding-them-296311.htmlhttps://gmatclub.com/forum/factors-multiples-hcf-lcm-296385.htmlIn today’s post, let us have a look at the different methods you can adopt to calculate HCF and LCM. We will also look at some important concepts on HCF and LCM towards the end of this post, which are useful tools to solve advanced questions on HCF and LCM.
The two most commonly used methods to find out HCF are :
• Prime Factorization
• Long Division
Let us take a few examples to understand each of the methods separately.
Prime Factorisation to find HCFConsider the numbers 30, 36 and 54. The first step is to prime factorise the numbers given. On prime factorizing the numbers, we have,
30 = 2 * 3 * 5
36 = \(2^2\) * \(3^2\)
54 = 2 * \(3^3\)
To find the HCF, list out all the prime factors and take the lowest power of each prime factor, considering all the numbers given. In our example, lowest power of 2 is 1, lowest power of 3 is also 1 and the lowest power of 5 is 0 – remember, 5 is not present in all the numbers given and therefore, you cannot say that it is a COMMON factor.
Therefore, HCF = \(2^1\) * \(3^1\) * \(5^0\) = 2 * 3 * 1 = 6.
Summarizing the prime factorization method of finding HCF, we have the following steps:
1. Prime factorise the given numbers and write them in the form of \(a^p\) * \(b^ q\) * \(c^r\) *……… where a, b, c….. etc., are the prime factors and p, q, r…… etc are the powers.
2. List out all the prime factors that you can see in the equations
3. Take the lowest power of each prime factor and multiply
4. The product obtained will represent the HCF.
Long Division method to find HCFThis method is preferred over the prime factorization method, when the numbers are slightly larger – bigger three digit numbers, probably. This is because, the time taken to prime factorise bigger three digit numbers is higher compared to smaller three- digit numbers and two-digit numbers.
This is not to say that you will always be required to prime factorise a number like, say 3969, on the GMAT, because, that’s not the objective of the test. However, you may be required to prime factorise numbers like 600 or 900 or 750 and so on.
So, let’s take two more numbers to learn this method. Consider the numbers 225 and 375.
1. The smaller number is always taken as the divisor and the larger number is taken as the dividend. The first step in the method is the same as normal division.
2. The divisor of the first step becomes the dividend of the second step and the remainder of the first step becomes the divisor of the second step.
3. This procedure is continued till we obtain a remainder of 0. The divisor of the last step of the division i.e. the divisor when the remainder became 0, represents the HCF.
We have demonstrated the process of finding the HCF of 225 and 375, in the figure below:
Attachment:
24th May 2019 - Post - Long Division.JPG [ 22.06 KiB | Viewed 5237 times ]
So, what if there are more than 2 numbers for which we have to calculate the HCF??
You can pick any two of the numbers given and find out their HCF. After doing this, you can take this HCF and the third number and repeat the process again to find out the HCF of the three numbers.
For example, if we have to find the HCF of 225, 375 and 525, we already know that 75 is the HCF of 225 and 375. Now, we need to find out the HCF of 75 and 525 by the long division method, which will finally be the HCF of all the three numbers.
As mentioned in our post on Prime numbers, learn the smart method of prime factorizing larger numbers, due to which you will be able to save time and energy. In case you need to know what we have covered in that post, here’s a link to the post:
https://gmatclub.com/forum/prime-numbers-properties-application-part-295990.html#p2276846Also, for smaller two-digit numbers, you should try and use your multiplication skills to obtain the prime factorized form, without actually prime factorizing them.
Let us now have a look at the methods we can use to find out the LCM. Again, the two most commonly used methods are,
• Prime Factorisation
• Division
Prime Factorisation method to find LCMLet’s consider the same numbers again - 30, 36 and 54. The first few steps remain the same viz., prime factorise each of the given numbers and write them in the prime factorized form.
Now, to find the LCM, we will take the highest powers of each prime factor. So, we have,
30 = 2 * 3 * 5
36 = \(2^2\) * \(3^2\)
54 = 2 * \(3^3\)
LCM = \(2^2\)* \(3^3\) * \(5^1\) = 4 * 27 * 5 = 540.
Summarizing the prime factorization method of finding LCM, we have the following steps:
1. Prime factorise the given numbers and write them in the form of \(a^p\) * \(b^ q\) * \(c^r\) *……… where a, b, c….. etc., are the prime factors and p, q, r…… etc are the powers.
2. List out all the prime factors that you can see in the equations
3. Take the highest power of each prime factor and multiply
4. The product obtained will represent the HCF.
Division Method of finding LCMThis method is more commonly used than the prime factorization method. In this method, we divide all the given numbers by a common prime number. This process is repeated till we obtain all the quotients as 1. The product of all the prime numbers, used as divisors, will represent the LCM.
Let’s take the same three numbers – 225, 375 and 525 and calculate the LCM using the Division method. Here’s a demonstration of how it has to be done:
Attachment:
24th May 2019 - Post - Division LCM.JPG [ 28.2 KiB | Viewed 5148 times ]
By now, we hope that you are clear about the basic concepts of HCF & LCM and also the methods of calculating them.
As promised , in the last part of this post, we would like to highlight some important points related to HCF and LCM, which can prove very useful in problems on HCF and LCM.
1. The LCM of a set of numbers will always be divisible by the HCF of the same set of numbers. In other words, HCF will also be a factor of the LCM.
Therefore, you can always write, LCM (Set of numbers) = HCF(Set of numbers) * k , where k is an integer.
2. When all the numbers in the given set are equal, their LCM = their HCF. Otherwise, the LCM of the given set will always be larger than their HCF.
For example, LCM (30, 30, 30) = 30 and HCF (30, 30, 30) = 30.
3. If the HCF of two numbers, a and b, is h, then, a = h * \(f_1\) and b = h * \(f_2\) where \(f_1\) and \(f_2\) are co-prime numbers.
4. For two numbers a and b, LCM (a, b) * HCF (a,b) = a * b.
5. The above rule is not necessarily true for more than two numbers.
6. HCF of fractions = HCF ( All numerators ) / LCM (All denominators)
7. LCM of fractions = LCM (All numerators) / HCF (All denominators)
We hope that you have found this series of posts on Factors & Multiples useful and informative. In our last part of this series of posts, tomorrow, we will quickly review the important points that we covered in the three parts and post some good questions based on Factors & Multiples.
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See you in the next post.
Thank you!
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Crackverbal Prep Team
www.crackverbal.com