Sunday, 3 April 2016

11. Quantitative Aptitude Free study material - HCF and LCM Concept

Aptitude Questions

HCF and LCM  
Factor:- A factor of a given number is every number that divides exactly into that number
      Example : Factors of 12 are 1,2,3,4,6,12

Prime factorization

If a natural number is expressed as the product of prime numbers, then the factorization of the number is called its prime (or complete) factorization.
A prime factorization of a natural number can be expressed in the exponential form.
For example:
(i) 90 = 2×3×3×5 = 2 × 2³×3
(ii) 420 = 2×2×3×5×7 = 23 ×32×5×7.

Common Factor: A number which exactly divides all the given numbers is called common factor of those numbers.

Example   Find all the common factors of 18 and 24.
Factors of 18 are 1, 2, 3, 6, 9,18.
Factors of 24 are 1, 2, 3, 4, 6, 8,12, 24
The common factors of 18  and 24  are 1, 2, 3 and 6

Highest Common Factor:  The greatest number that exactly divides all the given numbers is called Highest Common Factor (HCF). It is also called Greatest Common Divisor (GCD) or Highest Common Divisor (HCD )or Greatest Common Factor ( GCF)

To find HCF of Numbers:
Method 1:-
Prime Factorization method:  This method deals with the prime factors of numbers. It involves the following steps
Step1 : Express each one of the given numbers as the product of prime factors.
Step2 : Identify common prime factors
Step3 : Compute the product of all common prime factors, provided each common prime factor must appear in the prime factorization of all the given numbers.
The product of least powers of common prime factors gives HCF.
Example  :   Find the HCF of 18 ,12,24 .
                   Using Prime factorization method to Find HCF
                 18=2 x 3 X 3 => 18 = 2 x 32 
                12=2 x 2 x 3=> 24 =22x 3
                24=2 x 2 X 2 X 3 => 23 x 3
          H.C.F. of the given numbers = the product of common factors with least index
                           Then HCF= 2 x 3 =6

Find the H.C.F. of 144,270 and 630
.Solution
Using Prime factorization method
     72 = 2×2×2×3×3 = 2³×3²
  
  270 = 2×3×3×3×5 = 21×3³×51
    630 = 2×3×3×5× 7 = 21×3²×5×71
H.C.F. of the given numbers = the product of common factors with least index
   HCF = 21×3² = 2×3×3 = 18

Method 2
Division Method to Find HCF of two numbers
This method involves the following steps
Step 1 : Divide the larger number by the smaller one to obtain the remainder
Step 2 : If remainder is 0, the divisor is the required HCF
Step 3: If remainder is not zero, take this remainder as a divisor and the first divisor    as the dividend
Step 4 : Repeat the process till zero is obtained as a remainder.
                  The last divisor is the required HCF

Example: HCF OF 84,140     
Explanation : 84)140(1
                        84
                        56) 84 (1
                              56
                             28) 56(2
                                   56
                                    0
28 is the HCF of 84 and 140

Division Method to Find HCF of more than two numbers is find HCF of any two numbers as H1, and find HCF of H1 and the numbers and so on. The last HCF obtained is the HCF of all the given numbers.

Multiple: When we multiply a given whole number by any other whole number, the result is a multiple of that number.  For example, 5 is the first multiple of 5 (because 5 x 1 = 5), 10 is the second multiple of 5, and so on.
  Example  : Multiples of 12 = 12,24,36,48,60,72,..

Common Multiple: A number which is exactly divisible by all the given numbers is common multiple.
Example: Common multiple of 3, 5, 6, 8 is 120.

Least Common Multiple (LCM): Definition of LCM  The least number which is exactly divisible by all the given number is LCM.
             In other words , LCM of two or more numbers is the smallest number which is a multiple of each of the number.
        LCM is also known as lowest common dividend ( LCD)

To Find LCM of Given Numbers:
Method 1
Prime Factorization Method To Find the LCM: 
  Step 1 :  Express each number as a product of prime powers
  Step 2 : For each prime factor choose the prime power with largest exponent among all numbers.
                 The LCM is the product of the chosen prime powers

Example : Find the LCM of 12, 15, and 20
                12= 2 x 2 x 3=3 x 22  
                15=3 x 5=3 x 5
                20=2 x 2 x 5=2x 5
Here the prime factors appear in the given numbers are 2, 3, 5. 
Their highest power are 22, 3 and 5.
LCM=22´3´5=60

Method II: 
Division Method To Find LCM : Write all the given numbers in a now divide them by any one of prime numbers. 2, 3, 5, 7, 11 etc. which will divide at least two given numbers? Write down the quotients and the other undivided numbers in a row below the first. Repeat this process till you get a row of numbers, which are prime to one another.
                The product of all the divisors and the numbers in the last row is their LCM.

Ex:          12, 15, 20,25
                3|12       15      20 ,    25
                2|4         5        20 ,    25
                2|2         5        10,     25
                5|1         5          5,     25
                    1       1          1 ,     5

       LCM=2 x 2 x 3 x 5 x 5 = 3000

LCM of fractions :
   LCM of fractions =
LCM of Numerators / HCF of Denominators

Find the LCM of
2 / 3
,
5 / 6
,
7 / 9
LCM of fractions =
LCM of Numerators / HCF of Denominators
=
LCM of 2,5,7 / HCF of 3,6,9
=
70 / 3

HCF of fractions :    
HCF of fractions =
HCF of Numerators / LCM of Denominators

Find the HCF of
2 / 3
,
5 / 6
,
7 / 9
Explanation : HCF of fractions =
HCF of Numerators / LCM of Denominators
=
HCF of 2,5,7 / LCM of 3,6,9
=
1 / 18

Some Important Points :
1.  HCF of any two consecutive numbers is always  1
2. If all the given numbers are prime numbers, then their HCF is 1
3.HCF of relatively prime numbers or co primes is always 1
4.LCM of any two consecutive numbers is always equal to their product
5.If all the given numbers are prime numbers, then their LCM is equal to product of all given prime numbers
6. LCM of co primes or relatively prime numbers is equal to their product

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