CLASS-6
RELATION BETWEEN LCM & HCF

RELATION BETWEEN LCM & HCF -

The LCM (Least Common Multiple) and HCF (Highest Common Factor), also known as GCD (Greatest Common Divisor), are two fundamental concepts in number theory that are often used in various mathematical operations and problem-solving scenarios. They are related to each other in the following ways:

  1. Divisibility Relationship:- The HCF of two or more numbers is the largest number that divides all of them evenly. In contrast, the LCM of two or more numbers is the smallest multiple that is divisible by all of them. So, the HCF divides the LCM evenly.
  2. Product Relationship:- For any two positive integers a and b, their product is equal to the product of their HCF and LCM:- a * b = HCF (a, b) * LCM (a, b).
  3. Prime Factorization:- The HCF of two numbers can be found by taking the product of the common prime factors raised to the lowest powers in their prime factorization. The LCM can be found by taking the product of all prime factors, both common and non-common, raised to their highest powers.
  4. Relationship through Division:- If you have two positive integers a and b, you can express them as a = HCF (a, b) * x and b = HCF (a, b) * y, where x and y are coprime (have no common factors other than 1). Then, the LCM (a, b) = HCF (a, b) * x * y.
  5. General Relationship:- If you have three numbers a, b, and c, you can relate their HCF and LCM through the following equation: LCM (a, b, c) = (a * b * c) / HCF (a, b, c).
  6. Euclidean Algorithm:- The Euclidean algorithm, used to find the HCF of two numbers, can also be used to indirectly find the LCM. Once you have the HCF of two numbers, you can use the formula LCM (a, b) = (a * b) / HCF (a, b) to find their LCM.

The LCM (Least Common Multiple) and HCF (Highest Common Factor), also known as GCD (Greatest Common Divisor), are two fundamental concepts in number theory and arithmetic, often used in various mathematical computations.

  1. Definition:- LCM:- The LCM of two or more integers is the smallest positive integer that is divisible by all of the given integers without leaving a remainder. HCF/GCD:- The HCF or GCD of two or more integers is the largest positive integer that divides all of the given integers without leaving a remainder.
  2. Relationship:- For any two positive integers a and b, the product of their LCM and HCF is equal to the product of the original numbers:- a X b = LCM (a,b) X HCF (a,b). This relationship is a consequence of the fundamental theorem of arithmetic, which states that any positive integer can be uniquely represented as a product of prime numbers raised to certain powers.
  3. Example:- Let's take an example to illustrate the relationship: Consider a = 12 and b = 18.  LCM (12, 18) = 36, and HCF (12, 18) = 6.  Now, using the relationship: 12 X 18 = 36 X 6.This relationship holds true for any pair of positive integers.
  4. Applications:- LCM and HCF are used in various real-world scenarios, including problems involving fractions, ratios, and proportions.In algebraic manipulations, LCM and HCF are often used to simplify expressions and solve equations.They play a crucial role in number theory, cryptography, and algorithm design.


Relation Between HCF & LCM :-


                      ( 1st Number X 2nd Number )

1)    L.C.M  =  --------------------------------

                                  H.C.F

 

                       ( 1st Number X 2nd Number )

2)    H.C.F  =  --------------------------------

                                   L.C.M

 

3) Product of two numbers = Product of their H.C.F & L.C.M

    ( 1st Number X 2nd Number ) =  H.C.F  X  L.C.M    


Example.1) Find the LCM & HCF of 24 & 36. 

Ans.)

LCM

In above picture HCF and LCM of 24 & 36 are 12 & 72 respectively.

If A = 24, B = 36, LCM = 72, and HCF = 12.

Then as per the condition, we have to prove A X B = LCM X HCF

So, A X B = 24 X 36 = 864

and, LCM X HCF = 72 X 12 = 864

therefore, A X B = LCM X HCF = 864     (Proved)



Example.2) Find the LCM & HCF of 504 & 700, also justify the relation between LCM & HCF through diagram.

Ans.)

LCM

In summary, the LCM and HCF are closely related concepts that provide insights into the divisibility and common factors of numbers. They are often used together in various mathematical applications, including solving problems involving fractions, ratios, and proportions.