Mathematics has various different concepts and LCM and HCF are amongst the most important topics amongst them. Thus, it is very important to build a strong foundation when it comes to these concepts. The full form of LCM is Least Common Multiple and HCF is Highest Common Factor. LCM defines the least number which divides exactly by two or more numbers whereas HCF defines the greatest factor present in between given two numbers.
HCF is also known as Greatest Common Factor (GCF) in some cases and LCM is also called Least Common Divisor in some cases. There are two prominent ways to find H.C.F and L.C.M, and these are the division method or Prime Factorization Method. The shorter method to find both HCF and LCM is the division method. The factor tree method is also an easy way to find HCF.
1.1 HCF Definition
The largest positive integer which divides the numbers without leaving a remainder happens to be the greatest common divisor or the gcd of two or more positive integers and is also called the Highest Common Factor (HCF). For example, let’s take 36 and 45. The highest common number which divides both 36 and 45 is 9 so the HCF f 36 and 45 will be 9.
1.2 LCM Definition
The smallest or the least positive integer or the least common multiple that divides both numbers is called an LCM. For example, let’s take 4 and 6, now if we take out multiples of both the numbers, the least common multiple that would occur in both the cases is 12 and thus 12 is the LCM of 4 and 6.
The formula which involves the derivation of both HCF and LCM is:
Product of two numbers = (HCF of two numbers) x (LCM of the two numbers).
Let’s say Z and E and the two numbers then after applying the formula we will get:
Z x E = H.C.F(Z,E) x L.C.M(Z,E).
The formula can also be edited to suit the need to take out H.C.F and L.C.M, such as:
H.C.F of two number= Product of two numbers/ L.C.M of two numbers.
L.C.M of two numbers = Product of Two numbers/H.C.F. of Two numbers.
Methods of Finding H.C.F and L.C.M
Here are the methods we can use to find the HCF and LCM of given numbers.
- Division method
- Prime factorization method.
We will learn about both of them one by one.
3.1 Division Method for HCF
The steps to find H.C.F of a given number through the division method are as follows:
1) Larger number/ Smaller Number
2) The divisor of the above step / Remainder
3) The divisor of step 2 / remainder. Keep doing this step till R = 0(Zero).
4) The last step’s divisor will be HCF.
3.2 Prime Factorization method for HCF
Take an example of finding the highest common factor of 144, 104, and 160.
Now let us write the prime factors of 144, 104, and 160.
144 = 2 × 2 × 2 × 2 × 3 × 3
104 = 2 × 2 × 2 × 13
160 = 2 × 2 × 2 × 2 × 2 × 5
The common factors of 144, 104 and 160 are 2 × 2 × 2 = 8
Therefore, HCF (144, 104, 160) = 8.
3.3 Division Method for LCM
We follow the following steps:
- We keep both the numbers in a table and keep dividing them by the smallest number which divides both of them.
- After getting all the numbers we multiply all of them to find the LCM.
For example, let’s take 60 and 45.
Now the LCM of 60 and 45= 2x2x3x3x5= 180.
3.4 Prime Factorization method for LCM
Step 1: Firstly, you need to find all the prime factors of each of the given numbers and write all the prime factors in exponent form.
Step 2: Now you need to list all the prime numbers found, using the highest exponent found for each of them.
Step 3: Lastly, multiply the list of prime factors you have with exponents together to find the Least common multiple.
For example: Find the LCM (12,18,30)
List down the prime factors of 12 = 2 × 2 × 3
The prime factors of 18 are = 2 × 3 × 3
List down the prime factors of 30 = 2 × 3 × 5
You need to list all the prime numbers found, the number of times as they occur most often for anyone given number and you need to multiply them together to find the Least common multiple.
After multiplying, 2 × 2 × 3 ×3 ×5 = 180
Using the concept of exponents instead, multiply together each of the prime numbers with the highest power.
So, the LCM (12,18,30) = 180.
4. Difference between HCF and LCM
|Highest Common Factor||Least Common Factor|
|The highest common factor is the largest positive integer which divides the numbers without leaving a remainder happens to be the greatest common divisor or the gcd of two or more positive integers.||The Least Common Factor is the smallest or the least positive integer or the least common multiple that divides both numbers.|
|HCF helps us in arranging something into rows and groups.||LCM helps us to figure when something will happen again simultaneously.|