Properties of HCF and LCM

Understanding the terms multiples and factors is essential for a deeper understanding of the definitions of LCM (Lowest Common Multiple) and HCF (Highest Common Factor). The Prime factorization method and the division method are two important methods for determining H.C.F. and L.C.M. Both techniques were taught in previous classes. The shortcut method to find both H.C.F. and L.C.M. is the division method. In this article, we will learn LCM, HCF meaning, and the relation between HCF and LCM of natural numbers.

The HCF is the highest common factor between two numbers or more. Whereas the LCM is the lowest common multiple which is exactly divided by these two numbers. HCF can also be called the Greatest Common Factor between any two numbers. LCM is also called Least Common Divisor between two or more numbers.

The two most common methods to calculate HCF and LCM are the prime factorization method and the division method. While the division method is the most common method to calculate HCF and LCM. Here we will talk about HCF and LCM.

What are HCF and LCM?

The full form of LCM is the lowest common factor, whereas HCF stands for Highest Common Factor. The largest factor of two or more numbers is called their HCF, whereas the least number that is exactly divisible by two or more numbers is defined by the L.C.M.

We can find the LCM and HCF of two or more numbers using two techniques using the division method and prime factorization method. However, the division method is considered the fastest way to find the HCF and LCM. We are discussing in this article the HCF and LCM in detail.

What is HCF?

The full form of HCF is the Highest Common Factor. HCF is also called the Greatest Common Factor or GCD (Greatest Common Divisor). The HCF of a subset of whole numbers is the largest positive integer that divides all the given numbers and leaves the remainder 0.

Factors and Multiples

Factors are all the numbers that divide a number entirely, that is, without leaving any remainder. For example, the number 24 can be divided evenly by 1, 2, 3, 4, 6, 8, 12, and 24. All of these numbers are factors of 24, and each of these numbers is a multiple of 24.

What is a Factor?

The factors of a number are the numbers that divide it exactly and leave a remainder 0. In other words, they are the exact divisors of the given numbers.

For example, the factors of 20 are 1,2,4,5,10 and 20.

Let us now understand the concept of HCF with an example:

Consider two numbers 25 and 20 for HCF, 

So,  we can see that:

Factors of  25 = 1,5,25

Factors of 20= 1,2,4,5,10,20

Here the factors of 25 are 1,5 and 25. While factors of 20 are 1,2,4,5,10,20. Here we can say that 1 and 5 are common factors of both of the numbers. Here the 5 is the highest common factor that is HCF. 

Hence the 5 is the highest common factor (HCF) of 20 and 25.

What is LCM?

The lowest common multiple is the full name of LCM. The LCM of two or more numbers is the number that is exactly divisible by all of them. 

Multiples: Multiples of a number are the numbers that we get by multiplying them with any of the numbers.

For example: The first 10 multiples of 3 will be 3,9,12,15,18,21,24,27 and 30. 

Now let us understand how to find LCM.

Consider two numbers 9 and 12

Multiples of 9 will be:

9* 1 = 9 

9*2 = 18

9 * 3 = 27

9 * 4 = 36

9*5 = 45 

Multiples of 12 will be 

12*1 = 12 

12*2 = 24

12* 3 = 36

12* 4 = 48

12* 5 = 60

Now here we can see that the lowest multiple which is the common multiple of 9 and 12 is 36. Hence LCM of the 9 and 12 is 36. 

LCM and HCF Formula

The following is the formula that shows the relation between  HCF and LCM:

Product of Two numbers = (HCF of the two numbers) x (LCM of the two numbers)

If A and B are two numbers, then the product of HCF and LCM of two numbers equals the product of the two numbers, according to the formula.

H.C.F.(A,B) x L.C.M.(A,B) = A x B 

We can also write LCM and HCF formulas in terms of HCF and LCM, such as:

HCF of two numbers = \[\frac{\text{Product of two numbers}}{\text{LCM of two numbers}}\]

LCM of two numbers= \[\frac{\text{Product of two numbers}}{\text{HCF of two numbers}}\]

Now we will understand the full form of HCF and LCM. The full form of H.C.F. is the Highest Common factor and that of L.C.M. is Least Common Multiple.

Properties of HCF and LCM are given Below:

Property 1: The product of the LCM and HCF of any two natural numbers is the same as the product of the numbers themselves.

HCF x LCM = Product of the Numbers 

Let's consider A and B as two numbers.

LCM (A and B) × HCF (A and B) = A × B

Example: If 3 and 8 are two numbers.

First, let us find the LCM and HCF of 3 and 8 to verify the above property,

LCM (3,8) = 24

HCF (3,8) = 1

So, product of HCF and LCM= 24 x 1 =24

Also, 3 x 8 = 24

Hence, proved.

Note: We can apply this property for only two numbers.

Property 2: HCF of co-prime numbers is 1. As a result, the product of the numbers is equal to the LCM of the given co-prime numbers.

Product of the Numbers = LCM of Co-prime Numbers

Consider the numbers 21 and 22 as an example of coprime numbers.

LCM of 21 and 22 = 462

Product of 21 and 22 = 462

LCM (21, 22) = 21 x 22

Example: Find the LCM and HCF of two fractions 4/9 and 6/21.

Given, 4 and 6 are the numerators. 9 and 12 are the denominators.

LCM (4, 6) = 12

HCF (4, 6) = 2

LCM (9, 21) = 63

HCF (9, 21) = 3

Now as per the formula, we can write:

LCM (4/9, 6/21) = 12/3 = 4

HCF (4/9, 6/21) = 2/63

Property 3: HCF of any two or more numbers will never be greater than any of the given numbers.

Consider two numbers 4 and 8.

HCF(4,8)=4

One of the numbers is 4 and the other is 8, which is greater than HCF (4, 8).

Property 4: The LCM of any two or more numbers is never less than any of the numbers issued.

For instance, the LCM of 4 and 8 is 8, which is not smaller than either of them.

LCM of Two Numbers

The smallest common multiple (SCM) of two numbers is a positive integer that is divisible by both numbers. The least common multiple (LCM) is the number that divides two or more numbers evenly. If a and b's LCM is the same as c, then c should be equally divisible by both a and b.

Example: Let's say we need to find the LCM of two numbers, 8 and 12. Let's see how many times these two numbers are multiplied.

Multiples of 8 are 16, 24, 32, 40, 48, 56 and so on  …

Multiples of 12 are 24, 36, 48, 60, 72, 84 and so on…

We can see, the least common multiple or the smallest common multiple of two numbers, 8 and 12 is 24.

HCF and LCM Examples

1. Prove that LCM(9 and 12) x HCF(9 and 12) = Product of 9 and 12 

Solution: First we will factorize 9 and 12:

9 = 3 x 3 = 3²

12 = 2 x 2 x 3 = 2²

LCM of 9 and 12 = 2² x 3² = 4 x 9 = 36  

3 is common between multiples of 9 and 12. Hence HCF of 9 and 12 is 3.

LCM(9 and 12) x HCF(9 and 12) = 36 x 3

Product of 9 and 12 = 9 x 12

Hence proved!

2. Given 8 and 9 are Two Co-prime Numbers. Using these Numbers, Verify LCM of Co-prime Numbers = Product of the Numbers.

Solution: First, we will find LCM and HCF of 8 and 9:

8 = 2 x 2 x 2 = 2³

9 = 3 x 3 = 3²

LCM of 8 and 9 = 2³ x 3² = 8 x 9

As 1 is a common multiple of 8 and 9. Hence, the HCF of 8 and 9 is 1.

Product of both numbers i.e 8 and 9 = 8 x 9 =72

Hence, LCM of co-prime numbers = Product of the numbers.

Therefore, verified!

3. Find the HCF of \[\frac{12}{25}\], \[\frac{9}{10}\], \[\frac{18}{35}\], \[\frac{21}{40}\].

Solution: Find prime factors of all numbers:

12 = 2 x 2 x 3

9 = 3 x 3

18 = 2 x 3 x 3

21 = 3 x 7

HCF(12, 9, 18, 21 ) = 3

Now, factor all denominators values to find LCM:

25 = 5 x 5

10 = 2 x 5

35 = 5 x 7

40 = 2 x 2 x 2 x 5

LCM(15, 10, 35, 40) = 5 x 5 x 2 x 2 x 2 x 7 = 1400 

Therefore HCF \[\frac{12}{25}\], \[\frac{9}{10}\], \[\frac{18}{35}\], \[\frac{21}{40}\] = \[\frac{HCF(12, 9, 18, 21)}{LCM(25, 10, 35, 40)}\] = \[\frac{3}{1400}\]

Conclusion

The product of two numbers' highest common factor (H.C.F.) and lowest common multiple (L.C.M.) is equal to the product of two numbers, according to the above explanations. All of the properties of HCF and LCM have been discussed, along with solved examples.