LCM and HCF or GCD Explained: Factors, Multiples, Prime Factorization, & Fastest Methods

LCM and HCF are two of the most important ideas in number theory. They help you understand how numbers are connected, how they share factors, how they fit into groups, and how to solve problems faster. Many students first meet LCM and HCF as school topics, but these ideas are much bigger than that. They are used in fractions, ratios, time problems, scheduling, measurement, algebra, and competitive exams.

LCM stands for Least Common Multiple. It tells us the smallest number that two or more numbers can divide into exactly. HCF stands for Highest Common Factor. It tells us the biggest number that divides two or more numbers exactly. In some countries and books, HCF is also called GCD, which means Greatest Common Divisor. They are the same idea.

If you understand LCM and HCF deeply, you can solve many math questions much faster. You can simplify fractions, compare quantities, solve word problems, and even see patterns in numbers more clearly.

In this article, you will learn everything about LCM and HCF from the absolute basics to advanced tricks, including factors, multiples, prime factorization, division method, ladder method, Euclid’s algorithm, and quick exam strategies.

What Are Factors and Multiples?

Before LCM and HCF, you must understand factors and multiples. They are the most essential part of them.

Factors

A factor is a number that divides another number exactly without leaving a remainder. In short, values that can divide a number completely are its factors.
Example:
Factors of 12 are: 1, 2, 3, 4, 6, 12.
Because:
12 ÷ 1 = 12
12 ÷ 2 = 6
12 ÷ 3 = 4
12 ÷ 4 = 3
12 ÷ 6 = 2
12 ÷ 12 = 1

For 12:
1 × 12 = 12
2 × 6 = 12
3 × 4 = 12
Just like that we do all other factors. Let’s learn more.

Multiples

A multiple is the result of multiplying a number by whole numbers. But what are whole numbers. Whole numbers are the set of natural numbers (counting numbers) combined with zero, including 1,2,3,4,5,6,7,.. and so on up to infinity. They are non-negative, complete numbers without any fractional or decimal parts.
Example: Multiples of 3 are.
3, 6, 9, 12, 15, 18, 21, 24, …
These are formed by:
3 × 1, 3 × 2, 3 × 3, 3 × 4, and so on.

Important Difference

Factors divide a number. Multiples are made by a number multiplication. This difference is very important for LCM and HCF.

What Is HCF / GCD?

HCF means Highest Common Factor. In some countries, it is called GCD. It is the greatest number that divides two or more numbers exactly.
Example: Find the HCF of 12 and 18. This is Listing Factors Method.
Factors of 12 are: 1, 2, 3, 4, 6, 12
Factors of 18 are: 1, 2, 3, 6, 9, 18
Common factors: 1, 2, 3, 6
Highest common factor: 6
So:
HCF of 12 and 18 = 6.

Do you get it? It is easy, right? But what if the values are too big. There are three methods to solve. We will learn more about them further.
Example: Find the HCF of 32 and 52.
Doing factors now takes a lot of time, so: Prime Factorization Method.
$\begin{array}{|c|l|r|} \hline Calculations & Factors & Values \\ \hline 32÷2 = 16 \\ 52÷2 = 26 & \bold{2} & \bold{32, 52} \\ \hline 16÷2 = 8 \\ 26÷2 = 13 & \bold{2} & 16, 26 \\ \hline 8÷2 = 4 & 2 & 8, 13 \\ \hline 4÷2 = 2 & 2 & 4, 13 \\ \hline 2÷2 = 1 & 2 & 2, 13 \\ \hline 13÷13 = 1 & 13 & 1, 13 \\ \hline \ \bold{HCF = 4} & ~ & 1, 1 \\ \hline \end{array}$
Highest Common Factors (HCF): 2 × 2 = 4.
Did you notice that on left side “Factors”, we always pick and multiply the most common shared factors in both values.

HCF as the Greatest Common Divisor

Some books and countries use GCD instead of HCF. The difference is the name only, nothing else.
HCF = Highest Common Factor
GCD = Greatest Common Divisor

They mean the same thing.
Example:
HCF of 24 and 36 = 12
GCD of 24 and 36 = 12

Think of HCF or GCD as: How big can the common part be?

What Is LCM?

LCM means Least Common Multiple or some call it Lowest Common Multiple. It is the smallest number that is a multiple of two or more numbers.
Example: Find the LCM of 4 and 6. We are using The Listing Multiple Method.
Multiples of 4 are: 4, 8, 12, 16, 20, 24, …
Multiples of 6 are: 6, 12, 18, 24, 30, …
Common multiples: 12, 24, …
Least / Lowest common multiple: 12
So:
LCM of 4 and 6 = 12.

Did you notice, how we pick only lowest factor? But what if the values are too big. There are three methods to solve. We will learn more about them further.
Example: Find the LCM of 32 and 52.
Doing Listing Multiple now takes a lot of time, so: Prime Factorization Method.
$\begin{array}{|c|l|r|} \hline Calculations & Factors & Values \\ \hline 32÷2 = 16 \\ 52÷2 = 26 & 2 & \bold{32, 52} \\ \hline 16÷2 = 8 \\ 26÷2 = 13 & 2 & 16, 26 \\ \hline 8÷2 = 4 & 2 & 8, 13 \\ \hline 4÷2 = 2 & 2 & 4, 13 \\ \hline 2÷2 = 1 & 2 & 2, 13 \\ \hline 13÷13 = 1 & 13 & 1, 13 \\ \hline \ \bold{LCM=2^5×13} & \bold{416} & 1, 1 \\ \hline \end{array}$
Least Common Factors (LCM): 2 × 2 × 2 × 2 × 2 × 13 = 416.
All we need to do is to multiply all common factors. That is it.

Think of LCM as: How small can the common meeting point be?

Listing Method: How to solve HCF and LCM

Do you remember? We have studied about Factors and Multiples. Now, you know why did we learn them and how and where to use them. What is the difference in using them.

Listing Factors Method For HCF

Yes, HCF uses factors for listing. In simple words, we never go ahead of the given values in question.
1st Example: Find HCF of 18 and 25.
Factors of 18 are: 1, 2, 3, 6, 9, 18
Factors of 25 are: 1, 5, 25
Common factors: 1
Highest common factor: 1
So:
HCF of 18 and 25 = 1.

2nd Example: Find HCF of 15 and 50.
Factors of 15 are: 1, 3, 5
Factors of 50 are: 1, 2, 5, 10, 25, 50
Common factors: 1, 5
Highest common factor: 5
So:
HCF of 15 and 50 = 5.

3rd Example: Find HCF of 14, 28, and 49.
Factors of 14 are: 1, 2, 7, 14
Factors of 28 are: 1, 2, 4, 7, 14, 28
Factors of 49 are: 1, 7, 49
Common factors: 1, 7
Highest common factor: 7
So:
HCF of 14, 28, and 49 = 7.

4th Example: Find HCF of 8, 24, 36, and 42.
Factors of 8 are: 1, 2, 4, 8
Factors of 24 are: 1, 2, 3, 4, 6, 8, 12, 24
Factors of 36 are: 1, 2, 3, 4, 6, 9, 12, 18, 36
Factors of 42 are: 1, 2, 3, 6, 7, 14, 21, 42
Common factors: 1, 2
Highest common factor: 2
So:
HCF of 8, 24, 36, and 42 = 1.

Listing Multiple Method For LCM

LCM uses multiples for listing, not factors. In simple words, we always go ahead of the given values in question.
1st Example: Find LCM of 3 and 5
Multiples of 3 are: 3, 6, 9, 12, 15, 18, …
Multiples of 5 are: 5, 10, 15, 20, 25, …
First least / lowest common multiple: 15
So:
LCM = 15

2nd Example: Find LCM of 7 and 12
Multiples of 7 are: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91…
Multiples of 12 are: 12, 24, 36, 48, 60, 72, 84, 96…
First least / lowest common multiple: 84
So:
LCM = 84

3rd Example: Find LCM of 8, 12, and 20
Multiples of 8 are: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104, 112, 120, 128…
Multiples of 12 are: 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 132…
Multiples of 20 are: 20, 40, 60, 80, 100, 120, 140…
First least / lowest common multiple: 120
So:
LCM = 120

4th Example: Find LCM of 18, 22, 30, and 35
This method becomes too long and time-consuming when dealing with unusual or many values, we should go for other two methods then.
First least / lowest common multiple: 6930
So:
LCM = 6930

This method is simple, but it is not always fast for large numbers.

Prime Factorization Method: For HCF and LCM

Prime factorization means writing a number as a product of prime numbers. A number greater than 1 not divided by any smaller natural number except 1. In simple words, a number that is only divided by 1 and itself.
Prime Numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, and so on.
Example:
36 = 2 × 2 × 3 × 3 = 2² × 3²
48 = 2 × 2 × 2 × 2 × 3 = 2⁴ × 3

This method helps you find both LCM and HCF easily.

Highest Common Factor Using Prime Factorization

The factorization method is almost same for HCF and LCM, but for HCF, we only pick most common factors used for all values given in question.
1st Example: Find the HCF of 216 and 332.
For practical purpose, the boxed table is using Ladder Method.
$\begin{array}{|c|l|r|} \hline Calculations & Factors & Values \\ \hline 216÷2 = 108 \\ 332÷2 = 166 & \bold{2} & \bold{216, 332} \\ \hline 108÷2 = 54 \\ 166÷2 = 83 & \bold{2} & 108, 166 \\ \hline 54÷2 = 27 & 2 & 54, 83 \\ \hline 27÷3 = 9 & 3 & 27, 83 \\ \hline 9÷3 = 3 & 3 & 9, 83 \\ \hline 3÷3 = 1 & 3 & 3, 83 \\ \hline 83÷83 = 1 & 83 & 1, 83 \\ \hline \ \bold{HCF = 2^2} & \bold{4} & 1, 1 \\ \hline \end{array}$
Below is Prime Factorization Method:
216: 1 × 2 × 2 × 2 × 2 × 3 × 3
332: 1 × 2 × 2 × 83
Highest Common Factors (HCF): 1 × 2 × 2 = 4.
Notice that on left side “Calculations”, we always pick and multiply the most common shared factors in both values.

2nd Example: Find the HCF of 112, 135, and 160.
Ladder Method for better understanding.
$\begin{array}{|c|l|r|} \hline Calculations & Factors & Values \\ \hline 112÷2 = 56 \\ 160÷2 = 80 & 2 & \bold{112, 135, 160} \\ \hline 56÷2 = 28 \\ 80÷2 = 40 & 2 & 56, 135, 80 \\ \hline 28÷2 = 14 \\ 40÷2 = 20 & 2 & 28, 135, 40 \\ \hline 14÷2 = 7 \\ 20÷2 = 10 & 2 & 14, 135, 20 \\ \hline 10÷2 = 5 & 2 & 7, 135, 10 \\ \hline 135÷3 = 45 & 3 & 7, 135, 5 \\ \hline 45÷3 = 15 & 3 & 7, 45, 5 \\ \hline 15÷3 = 5 & 3 & 7, 15, 5 \\ \hline 5÷5 = 1 & 5 & 7, 5, 5 \\ \hline 7÷7 = 1 & 7 & 7, 1, 1 \\ \hline \ \bold{HCF = None} & \bold{1} & 1, 1, 1 \\ \hline \end{array}$
This is Prime Factorization Method:
112: 1 × 2 × 2 × 2 × 2 × 7
135: 1 × 3 × 3 × 3 × 5
16: 1 × 2 × 2 × 2 × 2 × 2 × 5
Highest Common Factors (HCF): 1.
Notice that on left side “Calculations”, we always pick and multiply the most common shared factors in all given values. There are three given 112, 135, 160. But on left side, we mostly see two pairs, but never three. This way, we cannot pick any factors besides 1. So the correct answer of HCF is 1.

3rd Example: Find the HCF of 105, 140, 175, and 210.
Ladder Method for better visual understanding.
$\begin{array}{|c|l|r|} \hline Calculations & Factors & Values \\ \hline 140÷2 = 70 \\ 210÷2 = 105 & 2 & \bold{105, 140, 175, 210} \\ \hline 70÷2 = 35 & 2 & 105, 70, 175, 105 \\ \hline 105÷3 = 35 \\ 105÷3 = 35 & 3 & 105, 35, 175, 105 \\ \hline 35÷5 = 7 \\ 35÷5 = 7 \\ 175÷5 = 35 \\ 35÷5 = 7 & \bold{5} & 35, 35, 175, 35 \\ \hline 35÷5 = 7 & 5 & 7, 7, 35, 7 \\ \hline 7÷7 = 1 \\ 7÷7 = 1 \\ 7÷7 = 1 \\ 7÷7 = 1 & \bold{7} & 7, 7, 7, 7 \\ \hline \ \bold{HCF = 5×7} & \bold{35} & 1, 1, 1, 1 \\ \hline \end{array}$
The Prime Factorization Method:
105: 1 × 3 × 5 × 7
140: 1 × 2 × 2 × 5 × 7
175: 1 × 5 × 5 × 7
210: 1 × 2 × 3 × 5 × 7
Highest Common Factors (HCF): 1 × 5 × 7 = 35.
Notice that on left side “Calculations”, we always pick and multiply the most common shared factors in all given values. Only 5 and 7 once divided all four given values, so we picked and multiplied them.

Least Common Factor Using Prime Factorization

In LCM, we pick all the factors and multiply them. That is it. Pick all and multiply. There is no need to have shared factors in all given values.
1st Example: Find the LCM of 216 and 332.
The Ladder Method for better understanding is given inside the container.
$\begin{array}{|c|l|r|} \hline Calculations & Factors & Values \\ \hline 216÷2 = 108 \\ 332÷2 = 166 & 2 & \bold{216, 332} \\ \hline 108÷2 = 54 \\ 166÷2 = 83 & 2 & 108, 166 \\ \hline 54÷2 = 27 & 2 & 54, 83 \\ \hline 27÷3 = 9 & 3 & 27, 83 \\ \hline 9÷3 = 3 & 3 & 9, 83 \\ \hline 3÷3 = 1 & 3 & 3, 83 \\ \hline 83÷83 = 1 & 83 & 1, 83 \\ \hline \ \bold{HCF = 2^3×3^3×83} & \bold{17,928} & 1, 1 \\ \hline \end{array}$
Below is the Prime Factorization Method:
216: 1 × 2 × 2 × 2 × 2 × 3 × 3
332: 1 × 2 × 2 × 83
Least Common Factors (LCM): 1 × 2 × 2 × 2 × 3 × 3 × 3 × 83 = 17,928.
The total common factors multiplication is the only difference between LCM and HCF. Look at the key pattern: 2 × 2 × 2 × 3 × 3 × 3 = 216.

2nd Example: Find the LCM of 14, 15, and 16.
Ladder Method:
$\begin{array}{|c|l|r|} \hline Calculations & Factors & Values \\ \hline 14÷2 = 7 \\ 16÷2 = 8 & 2 & \bold{14, 15, 16} \\ \hline 8÷2 = 4 & 2 & 7, 15, 8 \\ \hline 4÷2 = 2 & 2 & 7, 15, 4 \\ \hline 2÷2 = 1 & 2 & 7, 15, 2 \\ \hline 15÷3 = 5 & 3 & 7, 15, 1 \\ \hline 5÷5 = 1 & 5 & 7, 5, 1 \\ \hline 7÷7 = 1 & 7 & 7, 1, 1 \\ \hline \ \bold{LCM = 2^4×3×5×7} & \bold{1,680} & 1, 1, 1 \\ \hline \end{array}$
Prime Factorization Method:
14: 1 × 2 × 7
15: 1 × 3 × 5
16: 1 × 2 × 2 × 2 × 2
Lowest Common Factors (LCM): 1 × 2 × 2 × 2 × 2 × 3 × 5 × 7 = 1,680.
We multiplied all the common factors. Look at the tricky patter to save time: 2 × 2 × 2 × 2 = 16. The question value has two most common factors 14 and 16 which comes in table of 2. Look 14 + 2 = 16. There is no once trick, the real hack is that you solve many questions and get used to them then you will see the real tricks.

3rd Example: Find the LCM of 16, 19, 23, and 32.
The Ladder Method:
$\begin{array}{|c|l|r|} \hline Calculations & Factors & Values \\ \hline 16÷2 = 8 \\ 32÷2 = 16 & 2 & \bold{16, 19, 23, 32} \\ \hline 8÷2 = 4 \\ 16÷2 = 8 & 2 & 8, 19, 23, 16 \\ \hline 4÷2 = 2 \\ 8÷2 = 4 & 2 & 4, 19, 23, 8 \\ \hline 2÷2 = 1 \\ 4÷2 = 2 & 2 & 2, 19, 23, 4 \\ \hline 2÷2 = 1 & 2 & 1, 19, 23, 2 \\ \hline 19÷19 = 1 & 19 & 1, 19, 23, 1 \\ \hline 23÷23 = 1 & 23 & 1, 1, 23, 1 \\ \hline \ \bold{LCM = 2^5×19×23} & \bold{13,984} & 1, 1, 1, 1 \\ \hline \end{array}$
The Prime Factorization Method:
16: 1 × 2 × 2 × 2 × 2
19: 1 × 19
23: 1 × 23
32: 1 × 2 × 2 × 2 × 2 × 2
Lowest Common Factors (LCM): 1 × 2 × 2 × 2 × 2 × 2 × 19 × 23 = 13,984.
Multiply all common factors, that is it. There are many tricks behind it. If you learn them you can save your time, like notice how 2 × 2 × 2 × 2 × 2 is 32. Look the question, 16 and 32. At last we picked 32 × 19 × 23. When you learn doing Common Factor in your head. It all becomes so easy. The trick: 16 × 2 = 32.

Ladder Method for LCM and HCF

The ladder method is a fastest visual tool used in many classrooms. You have seen that up already, how smooth things go when we used Ladder method. The method is simple, put all values in one container and solve them.

I am going to pick all the examples given in Prime Factor here again but show you some concepts, tricks and hacks.

1st Example of HCF and LCM: 216 and 332.
$\begin{array}{|c|l|r|} \hline Calculations & Factors & Values \\ \hline 216÷2 = 108 \\ 332÷2 = 166 & \bold{2} & \bold{216, 332} \\ \hline 108÷2 = 54 \\ 166÷2 = 83 & \bold{2} & 108, 166 \\ \hline \ ~ & ~ & 54, 83 \\ \hline \end{array}$
I suggest you to check the 1st example of Prime Factorization in both LCM and HCF. Notice here, I did not solve it further. Because like I said, genius pros don’t solve it all. They know factors, tables, division, and multiplication all too well. It makes things very easy for them.
HCF: 2 × 2 = 4.
LCM: 2 × 2 × 54 × 83 = 17,928.
Get it, why we did not solve any further. Because after that the factor solves only one value at a time.

HCF: 2nd Example: Check the Prime Factorization above. There is no shortcut or trick for questions like that. You must solve it, or you are a pro who knows factors to well. That only can save your time.
HCF: 1
LCM: 30,240.

LCM: 2nd Example: 14, 15, and 16. This is exactly like the first example, but a few distinctions.
$\begin{array}{|c|l|r|} \hline Calculations & Factors & Values \\ \hline 14÷2 = 7 \\ 16÷2 = 8 & 2 & \bold{14, 15, 16} \\ \hline ~ & ~ & 7, 15, 8 \\ \hline \end{array}$
There is no common factor that used for all three values at once, so we can only pick 1.
HCF: 1.
LCM: 2 × 7 × 15 × 8 = 1,680.
I hope you are getting it. We are just saving time based on our knowledge of factors. It is like we see future. Compare the Full Table Method, I gave you on Prime Factorization 2nd Example of LCM and here.

The 3rd Example of HCF: 105, 140, 175, and 210. This is the perfect example of why there is no fixed trick, because you have to solve a question like that to fullest. I did not give the Ladder Table here again, check HCF 3rd example in Prime Factorization.
HCF: 5 × 7 = 35.
LCM: 2 × 2 × 3 × 5 × 5 × 7 = 2,100

3rd Example of LCM: 16, 19, 23, and 32. Another example that tells us to solve whole.
HCF: 1. Because there is no factors that used for all four values.
LCM: 13,984.

The ladder method is quick once you learn it. It is accurate as long as you solve it whole.

Division Method or Euclid’s Algorithm for HCF

This is a very popular and fast method. The both division and Euclid’s Algorithm are the same methods. There is no distinction. Let’s learn. But these methods are best when there are only two values given.
Example: Find HCF of 48 and 18
$\begin{array}{|c|l|r|} \hline Divisor & Dividend & Quotient \\ \hline \bold{18} & \bold{48} & 2 \\ \hline (18×2) & -36\ \ \ & ~ \\ \hline ~ & 12 & ~ \\ \hline \hline \bold{12} & \bold{18} & 1 \\ \hline (12×1) & -12\ \ \ & ~ \\ \hline ~ & 6 & ~ \\ \hline \hline \bold{6} & \bold{12} & 2 \\ \hline (6×2) & -12\ \ \ & ~ \\ \hline ~ & 0 & ~ \\ \hline \ \bold{HCF = 6} & ~ & ~ \\ \hline \end{array}$
Step 1: We do Division: Dividend ÷ Divisor
48 ÷ 18 = quotient 2 and remainder 12
18 ÷ 12 = quotient 1 and remainder 6
12 ÷ 6 = quotient 2 and remainder 0.
We calculate until we get remainder 0. The last divisor is 6.
So:
HCF = 6.
This is similar to Euclid’s algorithm.

Euclid’s algorithm:
Rule:
Keep dividing the larger number by the smaller number and continue with remainders until the remainder becomes 0. The last non-zero remainder is the HCF.
Example: Find HCF of 84 and 30
$\begin{array}{|c|l|r|} \hline Divisor & Dividend & Quotient \\ \hline \bold{30} & \bold{84} & 2 \\ \hline (30×2) & -60\ \ \ & ~ \\ \hline ~ & 24 & ~ \\ \hline \hline \bold{24} & \bold{30} & 1 \\ \hline (24×1) & -24\ \ \ & ~ \\ \hline ~ & 6 & ~ \\ \hline \hline \bold{6} & \bold{24} & 4 \\ \hline (6×4) & -24\ \ \ & ~ \\ \hline ~ & 0 & ~ \\ \hline \ \bold{HCF = 6} & ~ & ~ \\ \hline \end{array}$
Step 1: Division: Dividend ÷ Divisor
84 ÷ 30 = quotient 2 and remainder 24
30 ÷ 24 = quotient 1 and remainder 6
24 ÷ 6 = quotient 4 and remainder 0
So: The last divisor is 6.
HCF = 6.

Big Numbers HCF

Let’s try to solve 4 values at once using above methods.
Example: Find HCF of 32, 52, 66, and 80.
1st Step: Pick The Very First Two Values: 32 and 52.
2nd Step: The Bigger value ÷ Smaller value
Dividend ÷ Divisor = quotient and remainder
52 ÷ 32 = quotient 1 and remainder 20
32 ÷ 20 = quotient 1 and remainder 12
20 ÷ 12 = quotient 1 and remainder 8
12 ÷ 8 = quotient 1 and remainder 4
8 ÷ 4 = quotient 2 and remainder 0.
The last divisor is 4.
3rd Step: Lets pick third value.
66 ÷ 4 = quotient 16 and remainder 2
4 ÷ 2 = quotient 2 and remainder 0.
The last divisor is 2.
Last Step: Last value.
80 ÷ 2 = quotient 40 and remainder 0
The last divisor is 2.
HCF = 2. Final Answer.
This method is extremely efficient for big numbers.

Verify Your LCM and HCF Answer Is Correct Or Not?

The verification step requires both LCM and HCF, we know that the Ladder method is the fastest method to determine both at once. For two numbers: LCM × HCF = product of the numbers
Example: 12 and 18
LCM = 36
HCF = 6
Multiply both: 36 × 6 = 216
The Product Multiplication: 12 × 18 = 216. They match. This relationship is very useful for checking answers. But it works only when there are two values given. It will not work on three or more products / values.

Find LCM or HCF Using Product Numbers

It works only with two products / values. The method is the same like above. If you have studied the core concepts of Division and Multiplication, you know they are opposite of each.
Formulas:
LCM = Two products ÷ HCF
HCF = Two products ÷ LCM

Pick Above Example: 12 and 18.
Two Products: 12 × 18 = 216.
We know HCF: 6
Find LCM: 216 ÷ 6 = 36.

Now we know LCM is 36, so HCF will be: 216 ÷ 36 = 6.

LCM and HCF of Co-Prime Numbers

Two numbers are co-prime if their HCF is 1. Again, the trick works when there are two values / products given and their HCF is 1.
Examples:
8 and 15
7 and 20
9 and 10
If numbers are co-prime:
HCF = 1
Then:
LCM = product of the numbers
Example:
8 × 15 = 120 is the LCM.
7 × 20 = 140 is the LCM.
9 × 10 = 90 is the LCM.

HCF and LCM for Fractions

Fractions often need LCM and HCF. We need to do both in order to find correct answer from HCF or LCM of Fractions. It looks difficult but actually very easy, just takes time.

LCM of Fractions

We know a fraction means division value, and there are numerator and denominator.
Formula: LCM of fractions = LCM of numerators ÷ HCF of denominators
Example: 2/3 and 4/5
LCM of numerators: 2 and 4 = 4.
HCF of denominators: 3 and 5 = 1
LCM of both fractions is: 4/1 = 4

Another Example: LCM of 3/5, 4/9, and 6/15
Simplified third product / value: $\frac{6}{15} = \frac25$. Use the new third fraction.
LCM of numerators: 3, 4, and 2 = 12.
HCF of denominators: 5, 9, and 5 = 1.
LCM of fractions is: 12/1 = 12.

This topic is more advanced and appears in higher math. You can use as many values as you can, the method stays the same.

HCF of Fractions

The method is the same as above but opposites, we use. Let’s try to find out.
Formula: HCF of fractions = HCF of numerators ÷ LCM of denominators
Example: HCF of 2/3 and 4/5
HCF of numerators: 2 and 4 = 2
LCM of denominators: 3 and 5 = 15
So:
HCF = 2/15.

Another Example: HCF of 3/5, 4/9, and 6/15
Simplified third product / value: $\frac{6}{15} = \frac25$. Use the new third fraction.
HCF of numerators: 3, 4, and 2 = 1.
LCM of denominators: 5, 9, and 5 = 45.
HCF of fractions is: 1/45.

Word Problems in LCM and HCF

These topics often appear in real-life style questions.

1st Example: Scheduling
Two bells ring every 6 minutes and 8 minutes. After how many minutes will they ring together again?
Answer: Find LCM of 6 and 8.
Multiples:
6 → 6, 12, 18, 24, …
8 → 8, 16, 24, …
LCM = 24
So they ring together after 24 minutes.

2nd Example: Equal Cutting
A ribbon of 48 cm and another of 60 cm are to be cut into the largest equal pieces. What is the size of each piece?
Answer: Find HCF of 48 and 60.
48 = 2⁴ × 3
60 = 2² × 3 × 5
Common prime factors:
2² × 3 = 12
So each piece should be 12 cm long.

3rd Example: Minimum Time Together
One machine repeats every 12 seconds, another every 18 seconds. When will they repeat together?
Answer: Find LCM of 12 and 18
The LCM is 36.
So they repeat together after 36 seconds.

My Advice: Fast Tricks for HCF and LCM

Here are the best HCF shortcuts.

1. If one number divides the other, HCF is the smaller number.
2. Use prime factorization for exactness.
3. Use Euclid’s algorithm for large numbers.
4. If numbers end with the same divisor pattern, try divisibility rules first.
5. For small numbers, factor listing is often fastest.
6. If numbers are co-prime, HCF is 1.

For LCM

1. If one number divides the other, LCM is the larger number.
2. Use prime factorization for precise results.
3. For co-prime numbers, LCM is the product.
4. Use the formula LCM × HCF = product for two numbers.
5. Look for number patterns to avoid long listing.
6. Use multiples only when numbers are small.

Why LCM and HCF Matter

LCM and HCF are useful in:

• Simplifying fractions
• Adding and subtracting fractions
• Solving word problems
• Finding equal timing or scheduling
• Dividing items into equal groups
• Finding the biggest equal piece
• Ratios and proportions
• Algebra and number theory
• Competitive exam speed calculation

Common Mistakes in LCM and HCF

1. Confusing factors with multiples
2. Using lowest power for LCM by mistake
3. Using highest power for HCF by mistake
4. Forgetting that HCF is based on common factors only
5. Assuming LCM is always the product
6. Forgetting the formula works directly only for two numbers
7. Making errors in prime factorization

Example: For 6 and 8, LCM is not 14.
The correct LCM is 24.

When to Use Which Method

This is one of the most important thing when to use which method, we have learnt major three methods but Ladder Method is the one which can be used for all types of problems easily but if your goal is fastest solution then moving among methods is the best approach.

• Use factor listing for small numbers.
• Use prime factorization for medium numbers.
• Use Euclid’s algorithm for large numbers.
• Use the formula when one value is known.
• Use the ladder method when you want a visual fast technique.

This flexibility is what makes a student faster.

Real-Life Meaning of LCM and HCF

We always make jokes about Math that many things in math has no use in real life, but that is not true.

• HCF helps when you want the biggest equal piece.
• LCM helps when you want the next common meeting point.

Example:
Cutting many different size ropes into the largest equal pieces → HCF
Finding when two or more buses arrive together → LCM

This is the simplest way to remember the difference.

Conclusion

LCM and HCF are not just school formulas. They are powerful number tools that help you think faster and solve smarter. HCF shows the biggest common part. LCM shows the smallest common meeting point. Once you understand factors, multiples, prime factorization, Euclid’s algorithm, and the relationship between LCM and HCF, these topics become much easier and much more useful.

The best students do not memorize blindly. They look for structure, factor patterns, and shortcuts. That is how they solve problems faster and with more confidence.

Master these ideas, and you will be stronger in fractions, ratios, word problems, and advanced math.