Questions on HCF and LCM are the most common in the competitive examinations. We have already studied HCF & LCM in our elementary schools. Let’s recall from our memory.

# Highest Common Factor (HCF)

**Highest Common Factor (HCF) **or Greatest Common Divisor (GCD) is the greatest number that divides each of the given number exactly.

Let’s take an example. Suppose we need to find the HCF of 2 and 4. By definition, HCF is that number which divides both 2 and 4.

So, which is that number, which can divide both 2 and 4? By inspection, we know 1 and 2 can divide both the numbers.

However, 1 cannot be the HCF, as we need to choose the largest. So 2 is the HCF of 2 and 4. There are a number of techniques to find the HCF of 2 or more numbers. We will come to that in a while.

There are a number of techniques to find the HCF of 2 or more numbers.

## Technique 1 : Through Prime Factors

In this method, we represent each of the number through its prime factors. The product of all the factors which are common to all the numbers will be the HCF of the numbers. Let’s consider an example.

**Exercise 1 : **Find the HCF of 108, 288 & 360

**Solution :**

** **

**Step 1: **Represent each of the number in terms of its prime factors. So,

108 = 2^{2} x 3^{3}

288 = 2^{5} × 3^{2}

360 = 2^{3 }× 3^{2} × 5

**Step 2: **Select the factors which are common to all the numbers and multiply them.

Notice that the factors which are common to 108, 288, 360 is 2^{2} × 3^{2}

So, the HCF of 108, 288 and 360 is 2^{2} × 3^{2} = 36

Similarly, The HCF of 208 and 360 is 2^{3} × 3^{2} = 36

**Important : **This technique, in general takes a long time to find the HCF.

## Technique 2 : Through Division

Consider the same example. Find the HCF of 108, 288 & 360

If you have 2 or more numbers, instead of finding the HCF of all the numbers together, we can find the HCF of any 2 numbers. The HCF of these 2 numbers will be used to find the HCF of 3 numbers.

HCF(x, y, z) = HCF(h1, z)

where h1 is the HCF of x and y.

**Step 1 : **In the first step, we find the HCF of 108 and 288 using the division technique. The complete steps are illustrated in the following figure.

The principle behind division technique is to keep on dividing the divisor from the remainder till we get the remainder as 0. When the remainder is 0, the divisor used in the last division is the HCF of the 2 numbers.

So, the HCF of 108 and 288 is 36.

**Step 2 : **To find the HCF of 108, 288, and 360, it is sufficient to find the HCF of 36 and 360. We will use the division technique again. The complete steps are illustrated in the following figure.

The HCF of 36 and 360 is 36. So, the HCF of 108, 288, and 360 is 36.

**Example: **Find the HCF of 22, 54, 108 and 135.

**Solution: **HCF (22, 54, 108,135) = HCF (h1, h2)

Where h1= HCF (22, 54)

h2= HCF (108, 135)

**Step 1:** The steps for finding the HCF of 22 and 54 are illustrated in the following figure.

h1= HCF (22, 54) = 2

Now, notice an important thing. Once you have the HCF of 22 and 54 as 2, the overall HCF can be either 1 or 2 but cannot be more than 2.

By inspection, we can see, HCF of 2 and 135 would be 1 ( as 135 is an odd number)

So, the overall HCF would be 1. So, we can actually skip the rest of the steps.

**Step 2:** The steps in finding the HCF of 108 and 135 are illustrated in the following figure.

h2= HCF (108, 135) = 27

**Step 3 : **The overall HCF will be equivalent to the HCF of 2 and 27.

By inspection, we can tell, HCF of 2 and 27 would be 1. So, overall HCF is therefore 1.

**Exercise**

- Find the H.C.F. of 36 and 84.
- Find the H.C.F. of 2923 and 3239.
- The H.C.F. of 3556 and 3444.

Consider the previous question; let’s assume the same question would have been asked in the competitive examination. The question might appear something like this :

The HCF of 22, 54, 108 and 135.

- 27
- 1
- 22
- 36

After the completion of Step 1, you know that the HCF of 22 and 54 is 2. The overall HCF can never exceed 2. Now, looking at the options, you can easily elimination (1), (3) and (4). You are left with option (2) which by default is the right answer.

This technique is called “**Process of Elimination**”

# Least Common Multiple (LCM)

**Least Common Multiple (LCM)** is that least number which exactly divides each of the given number exactly.

Let’s take the same example. Suppose we need to find the LCM of 2 and 4. By definition, LCM is that number which is divisible by both 2 and 4.

Consider 4, 8, 12, and 16. Notice that 8, 12, 16 are divisible by both 2 and 4. However they cannot be considered LCM of 2 and 4. Since 4 is the smallest multiple of 2,4, 4 is the LCM of 2 and 4.

Let’s consider the previous example.

## Technique 1 : Through Prime Factors

**Exercise :** Find the LCM of 108, 288 & 360

**Solution : **

**Step 1: **Represent each of the number in terms of its prime factors. So,

108 = 2^{2} × 3^{3}

288 = 2^{5} × 3^{2}

360 = 2^{3 }× 3^{2} × 5

**Step 2 : **Select the factors product of highest powers of all the factors gives LCM

So, the LCM of 108, 288 and 360 is 2^{5} × 3^{3} × 5 = 4320

## Technique 2 : Factorization

**Example :** Find the LCM of 16, 24 and 36.

**Solution :** The steps for obtaining the LCM are illustrated in the following figure.

The idea is to factorize all the numbers using a common factor till we are left with only prime factors or just 1.

LCM = 2 x 2 x 2 x 3 x 2 x 3

= 2^{4} × 3^{2 }= 144

Irrespective of how big the number is, this technique is probably most straight forward and easier way to find the LCM of the numbers.

**Exercise**

- Find the L.C.M. of 144 and 185.
- Find the L.C.M. of 24, 36 and 40.
- Find the L.C.M. of 22, 54, 108, 135 and 198.

**Important : **The product of 2 numbers is equal to the product of their HCF and LCM.

# HCF and LCM of Fractions

**Example :** Find the H.C.F of 2/3, 8/9, 64/81 and 10/27.

**Solution :** = HCF(2, 8, 64, 10) / LCM(3, 9, 81, 27) = 2/81

**Example :** The HCF of 2 numbers is 11 and their LCM is 693. If one of the number is 77, find the other number.

**Solution : **

Product = HCF x LCM

77 x num = 11 x 693

num = 11 x 693 / 77 = 99

**Example :**** **Find the LCM of 25/6 and 15/4.

**Solution : **

LCM = LCM(25, 15) / HCF(6,4)

LCM = 75/2

HCF = HCF(25, 15) / LCM(6,4)

LCM = 5/12

**Example : **Find the HCF of 2^{2 }× 3^{3 }× 5^{5 }, 2^{3 }× 3^{2 }× 7and 2^{4 }× 3^{4 }× 5× 7^{2 }× 11.

**Solution : **Notice that the numbers are already represented as the product of their primes.

So, writing the numbers one below the other:

2^{2 }× 3^{3 }× 5^{5 }

2^{3 }× 3^{2 }× 7

2^{4 }× 3^{4 }× 5^{ }× 7^{2 }× 11

Now, select the factors which are common to all the numbers and multiply them

HCF = 2^{2 }× 3^{3 }× 5

In this tutorial on HCF and LCM, I have tried to cover all the typical question, you can expect in a competitive examination. If you have any doubts regarding HCF and LCM topic, please leave a comment. I will get back to you.

** **

Again, there is one typo in below line :-

2^5 × (2^3 )× 5 = 4320

It should be 3^3 inside the brackets :).

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Hey! writing an article over HCF and LCM, thats good..Intro …ok fine…but i have a doubt and a good assignment for your love for numbers..HCF*LCM = prod of numbers…does it hold true for any number of numbers..i mean prod of 3 numbers, 4 numbers…and how to find a particular number in case of 3 numbers a,b,c when their HCF(a,b,c) and LCM(a,b,c) are given?

Hey..

It is not possible. It is not a question of HCF or LCM. You basically have 3 unknowns and 2 equations. You cannot get numbers a, b and c.

I hope it clears your doubt.

Write to me on FB or if you are not clear with some points. I would be glad to help.

Thanks

Kiran

ghg

Note however that 15 x 2 and 15 x 3 x 2 would be a division of the factors as well. But that makes 30 and 90 – clearly the LCM is 90. Why?