Properties of HCF and LCM

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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 factorisation 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.


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.


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 formula in terms of HCF and LCM, such as:

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

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

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


Definition of LCM and HCF

We know that a number's factors are the exact divisors of that number. The highest common factor (H.C.F.) and the least common multiple (L.C.M.) will be discussed next (L.C.M.).

HCF meaning is Highest Common Factor. The greatest common divisor, or GCD, of two or more positive integers, is the largest positive integer that divides the numbers without leaving a remainder, according to mathematical rules. Consider the numbers 8 and 12. Since the maximum number that can divide both 8 and 12 is 4, the H.C.F. of 8 and 12 would be 4.

LCM (Least Common Multiple): In arithmetic, the least common multiple, or LCM, of two numbers, say a and b, is denoted by the symbol LCM (a,b). The LCM is the smallest or least positive integer divisible by both a and b. Take the positive integers 4 and 6 as an example.

Multiples of 4 are: 4,8,12,16,20,24…

Multiples of 6 are: 6,12,18,24….

From above, we can say, the common multiples of 4 and 6 are 12,24,36,48…and so on. In that set, the least common multiple is 12. Let's see if we can locate the LCM of 24 and 15.

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Hence, LCM of 24 and 15 will be the product of  2 × 2 × 2 × 3 × 5 which is 120.


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


Property 3 - LCM and HCF of Fractions:

LCM of fractions = \[\frac{\text{LCM of numerators}}{HCF of denominators}\]

HCF of fractions = \[\frac{\text{HCF of numerators}}{LCM of denominators}\]


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 4: 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 5: 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 absolutely 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 

Sol: First we will factorise 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 multiple in 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.

Sol: 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 common multiples of 8 and 9. Hence 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 \[\frac{12}{25}\], \[\frac{9}{10}\], \[\frac{18}{35}\], \[\frac{21}{40}\]

Sol: 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 value 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.

FAQs (Frequently Asked Questions)

1. How to Find LCM of Two Numbers?

Ans: There are four main methods to calculate the least common multiple of 2 numbers. These methods are:

  • Listing Multiples or Brute Force Method

  • Prime Factorization Method

  • Division Method or Ladder Method

  • GCD or GCF Method

2. What are the Methods to Find HCF and LCM?

Ans: Here are two methods for determining the HCF and LCM of numbers.

  • Prime factorisation method

  • Division method

3. Finding LCM By Division Method

Ans: We use the same approach we used to find the HCF using the division method to find the LCM of a given set of numbers using the long division method. The only difference will show up in the last step. We multiply all the numbers obtained at the end of the division instead of multiplying all the common factors on the left-hand side.

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