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Quantitative Aptitude :H.C.F and L.C.M SSC,IBPS, Bank, RRB,PSC Exams

H.C.F and L.C.M : Quantitative Aptitude for competitive (SSC,IBPS, Bank, RRB) Exams 

The concept of H.C.F and L.C.M is important from competitive exams point of view . In almost all competitive exams H.C.F and L.C.M questions are very common.

  1. Factors and Multiples:

If a number x divides another number y exactly, then we can say x is the factor of y. Here y is a multiple of x. 

Example: Factors of 15 are 1, 3, 5 and 15

Multiples of a number is its multiplication table itself.

Example: Multiples of 3 are 6, 9, 12, 15,18 …..


Highest Common Factor (H.C.F.) or Greatest Common Divisor (G.C.D.) or Greatest Common Measure(G.C.M):

The H.C.F. of two or more than two numbers is the greatest number that divides each of the numbers exactly.

There are two methods of finding the H.C.F. of given set of numbers:

  • Factorization Method: Express each one of given no  as the product of their prime factors & find  product of its least powers of common prime factors will give H.C.F.

Example :Find the HCF of 16,48,64
              16=24 , 48=24 ×3, 64=26
                   H.C.F=24 =16
  • Division Method: Divide the larger by the smaller one then; divide the divisor by the remainder. Repeat this process of dividing the preceding number by the remainder till zero is obtained as remainder. The last divisor is our H.C.F.

Example :H.C.F of 1134 and 1215
        ______
1134)  1215( 1
           1134           
                81)1134(14
                      1134 
                       000
                      
H.C.F of 1134 and 1215 is 81.      

Finding the H.C.F. of more than two numbers: Suppose we have to find the H.C.F. of three numbers, so, H.C.F. of [(H.C.F. of any two) and (the third number)] gives the H.C.F. of three given number. In the same way, we can get the H.C.F. of more than three numbers.

Example:H.C.F of 513,1134 and 1215
We already have H.C.F of 1134 and 1215.
H.C.F of 1134 and 1215 is 81.
So, required H.C.F.  is H.C.F. of 513 and 81
     ______
81 )513( 6
        486           
          27 )81(3
                 81 
                 00
H.C.F of given numbers=27

Least Common Multiple (L.C.M.):

Least number which is exactly divisible by each one of the given numbers is called their L.C.M.

There are two methods of calculating the L.C.M. of numbers:

  • Factorization Method: Resolve each one of the given numbers into the product of its prime factors. Then, a product of the highest powers of all the factors gives L.C.M.
  • Example :Find the LCM of 16,48,64
                  16=24 , 48=24 ×3, 64=26
                       L.C.M=26 ×3 =192
  • Common Division Method (short-cut): Arrange the given numbers in a row in any order .Divide them by the smallest number which divides at least two of the given numbers exactly and carries forward the numbers which are not divisible. Repeat the same process till no number is further divisible except no 1. The product of the divisors and the undivided numbers is the L.C.M. of the given numbers.

 Example:Find the LCM of 16,24,36 and 54

2

16  -   24   - 36 -  54

2

8   -    12  -  18 -   27

2

4   -      6   -   9 -   27

3

2  -       3  -    9 -   27

3

2  -       1 -     3  -   9

 

2  -       1 -     1  -   3

L.C.M=2×2×2×3×3×2×3=432

L.C.M and H.C.F important Facts and formulae

  • A product of H.C.F. and L.C.M = Product of two numbers.
     Example: HCF of 12 and 16 =4, LCM of 12 and 16=48 ,product of no=12*16=     192 
           Product of LCM and HCF=48*4=192
  • Co-primes: Co-primes are set of two numbers whose H.C.F. is 1.
     Example :(2,3), (3,5), (17,19), (21,22), (29,31), (41,43), (59,61), (71,73),         (99,100)
     Any pair of two consecutive numbers will be co prime and set of any two                prime numbers  will                also be co prime
  • H.C.F. and L.C.M. of Fractions:

            1. H.C.F. = H.C.F. of Numerators/L.C.M. of Denominators

             Example: The H.C.F of 9/10, 12/25, 18/35, 21/40 is:

             H.C.F of given fractions  = (H.C.F of 9, 12, 18, 21 )/(L.C.M of 10, 25, 35, 40) =                      3/1400

            2. L.C.M. = L.C.M. of Numerators/H.C.F. of Denominators

             Example: Find the LCM of 2/3, 8/9, 16/81, and 10/27

             LCM of given fraction = (LCM of 2,8,16,10)/(HCF of 3,9,81,27) =80/3 

  • H.C.F. and L.C.M. of Decimal numbers:
    • H.C.F and L.C.M of decimal numbers can be calculated by converting the decimal numbers into fractions & then following the same approach of finding the H.C.F and L.C.M of fractions as given above.

      • Example: Find the the HCF and LCM of 0.63 ,1.05 and 2.1
      Making the same number of decimal places ,the given numbers are 0.63, 1.05 and 2.10
      Without decimal places , these numbers are 63,105 and 210.
      Now, H.C.F of 63,105 and 210 is 21
      LCM of 63,105 and 210 is 630
      LCM of 0.63 ,1.05 and 2.1

  • HCF and LCM of decimal fractions :In given numbers ,make the same number of decimal places by annexing zeros in some numbers, if necessary .Considering these numbers without decimal points ,Find H.C.F and L.C.M as the case may be .Now ,in the result ,mark off as many decimal places as there in each of the given numbers. 
  • Comparison of Fractions :To compare two fractions ,find the L.C.M. of denominators. Convert the each of the fractions into an equivalent fractions with L.C.M. as the denominator, by multiplying both the numerator and denominator by the same number. The resultant  

 

Let us now discuss some questions based on this concept:

Quantitative Aptitude for practice papers on H.C.F and L.C.M for Banking ,SSC,RRB,PSC,Insurance exams.

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