Properties of HCF (GCD) & LCM

To understand the concept of HCF (Highest Common Factor) / GCD (Greatest Common Divisor) & LCM (Lowest Common Multiple) in completely, we have to recall the terms Factors and Multiples. 

We may also recall Least Common Multiple ( LCM) and Highest Common Factor (HCF) or Greatest Common Divisor (GCD) before going through their properties.

Least Common Multiple ( LCM)

The least common multiple ( LCM ) is also referred to as the lowest common multiple or smallest common multiple of two integers. For any two integers a and b, usually denoted by LCM(a, b), is the smallest positive integer that is divisible by both a and b.

For example, The  L.C.M of 3 and 4 is 12.

Highest Common Factor (HCF) Or Greatest Common Divisor (GCD / gcd) 

The Highest Common Factor ( HCF) or greatest common divisor (GCD / gcd) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers.
For any two integers a and b, usually denoted by HCF (a, b), is the largest integer that can divide both  numbers a and b. (a , b) can be also written as for simplicity. 
For example, The  H.C.F of 10 and 15 is 5.

Important Properties of HCF and LCM

• 1) The Highest Common Factor ( HCF) or greatest common divisor (GCD) of two or more integers is always LESS  than  to the  given numbers. 

For example, The  H.C.F of 10 and 15 is 5.                                 

If we consider another example,The  H.C.F of 14 and 28 is 14.

• 2) The Least Common Multiple  LCM)  of two or more integers is always GREATER than  or EQUALS to  given numbers.                                                       

For example, The  L.C.M of 10 and 20 is 20.  

If we consider another example,The  L.C.M of 10, 15 and 20 is 60.

• 3)  Relation between LCM and HCF (Formula for finding HCF and LCM)

Given relations is very popular in mathematics to find either LCM (if HCF is known) or HCF (if LCM is known)  or both are known along with one number then we can find another unknown number.

'The product of Least Common Multiple( LCM) and Highest Common Factor (HCF) or Greatest Common Divisor (GCD) of any two Numbers/ Polynomials is equaled to their Products'. 

in other words, mathematically we can write as:-

                    LCM × HCF = Product of the Numbers

if a and b are two numbers, then.

                    HCF (a , b) x LCM (a , b)   = a × b.

Above results  can be used directly or summarized as in different situations

LCM (a , b)   = (a × b) / HCF (a , b).

HCF (a , b) = (a × b) / LCM (a , b).

a ( one number) = LCM (a , b) × HCF (a , b) / b 

Illustrative Examples for this third property / Formula / Results.

Example 1:- The LCM and HCF of two numbers are 150 and 4 respectively. If  25 is a number given then find the other number.

Solution: As we know  that the product of two numbers is equal to the product of HCF and LCM of two numbers & HCF and LCM are given to us. Also, one of the numbers is given to us. Thus, we need to find the other number so we may use the above relation.

So,  HCF x LCM (or LCM x HCF )  = Product of the Numbers

=> 150 × 4 = 25 × x

=> x = 24

Therefore  required answer is 24.

OR

We can use direct relation to verify the above results

a ( one number) = LCM (a , b) × HCF (a , b) / b 

a = (150 x 4 ) / 25

a = 24.

Therefore  required answer is 24.

Example 2:-  Prove the above formula 

 LCM (18 & 12) × HCF (18 & 12) = Product of 18 and 12

Solution:- First of all we have to find the prime factorisation of given Numbers 18 and 12

18 = 2 x3 x 3 =  2 x 3²

12 = 2 x 2 x 3 = 2² x 3 

Now we have to calculate their LCM and HCF

LCM of 18 and 12 = 2² × 3² = 4 × 9 = 36

HCF of 18 and 12 = 3 × 2 = 6

LHS (Left Hand Side) =

LCM (18 & 12) × HCF (18 & 12) = 36 × 6 = 216

RHS (Right Hand Side) = 

Product of 18 and 12 = 18 × 12 = 216

Now 

LHS (Left Hand Side) =  RHS (Right Hand Side) = 216

Hence, LCM (18 & 12) × HCF (18 & 12) =  18 × 12 = 216

• 4)  Formula for finding HCF and LCM of Fraction is such as 

L.C.M. = L.C.M. of Numerator / H.C.F. of Denominator

H.C.F. = H.C.F. of Numerator / L.C.M. of Denominator 

Now we solve some problems for finding LCM and HCF of Fractions:-

CFofnumeratorsLCMofdenominators

Example 3:-  Find L.C.M. of 

                     4 / 3, 8 / 81, 64 / 9 ,10 / 27 

Solution: 

Before using this formula we have  to find LCM of Numerator and HCF of Denominator.

L.C.M. =	L.C.M. of Numerator
	H.C.F. of Denominator

L.C.M. of Numerators = 4, 8, 64, 10 ( By Prime Factorisation)

4 = 2²
8 = 2³
64 = 2⁶
10 = 2 × 5
L.C.M of 4, 8, 64, 10 = 2⁶ × 5 = 320

H.C.F. of Denominators = 3, 81, 9, 27  ( By Prime Factorisation)
3 = 3¹
81 = 3⁴
9 = 3²
27 = 3³
H.C.F. of 3, 81, 9, 27 = 3

L.C.M. of 

                     4 / 3, 8 / 81, 64 / 9 ,10 / 27  = 320 / 3

Therefore 320/ 3 is our required Answer.

We will solve one more Examples

Example 4:- Find the HCF of 

415, 910, 1835, 2130

Solution: Before using this formula we  have to find LCM of Denominator and HCF of Numerator.

H.C.F. =	H.C.F. of Numerator
	L.C.M. of Denominator

H.C.F. of Numerators = 4, 9, 18, 21  ( By Prime Factorisation) 

4 = 2 × 2 
9 = 3 × 3
18 = 2 × 3 × 3
21 = 3 × 7
HCF (4, 9, 18, 21) = 1
L.C.M. of Denominators = 15,10,35, 30 ( By Prime Factorisation)

15 = 3 × 5
10 = 2 × 5
35 = 5 × 7
30 = 2 × 3 × 5
LCM(15, 10, 35, 30) = 5 × 3× 2 ×  7 = 210
The required 

HCF = HCF(4, 9, 18, 21)/LCM(4, 9, 18, 21) = 1/210

Therefore 1 / 210  is our required Answer.

Example 5:- Find the HCF of 

415, 1681, 89.

Solution: Before using this formula we have to find LCM of Denominator and HCF of Numerator.

H.C.F. =	H.C.F. of Numerator
	L.C.M. of Denominator

OR                H.C.F. =	H.C.F. of 4, 16, 8
	L.C.M. of 15, 81, 9

H.C.F. of Numerators = 4, 16, 8  ( By Prime Factorisation)

4= 2²
16 = 2⁴

8 = 2³
Number with least index = 2² = 4
H.C.F. of 4, 16, 8 = 4

L.C.M. of Denominators = 15,81,9 ( By Prime Factorisation)

15= 3 x 5¹
9 = 3²
81 = 3⁴
Number with highest index =  5 x 3⁴  = 5 x 81 =405
L.C.M. of 15, 81, 9 = 405

H.C.F. =	4
	405

Therefore 4 / 405  is our required Answer.

• 5 ) Since by the definition of the co-prime number we know that "HCF of co-prime numbers is 1". Therefore LCM of given co-prime numbers is equal to the product of the co-prime numbers.

Co-prime Numbers LCM  = Product Of The Numbers

Example 6:-  Prove the above formula 

 LCM of Co-prime Numbers 5&7 = Product Of The Numbers 5&7

Solution:- 

LHS (Left Hand Side) =
      5= 5¹                  7 = 7¹

LCM (5 & 7) = 5 × 7 = 35

RHS (Right Hand Side) = 

Product of 5 and 7 = 5× 7 = 35

Now 

LHS (Left Hand Side) =  RHS (Right Hand Side) = 35.

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