H.C.F and L.C.M Questions - 1

Find the HCF of 84, 280 and 364
  • 84
  • 36
  • 28
  • 42
  • 21
Explanation   

HCF is the highest common factor which divides all the given numbers exactly.


280 )
364
( 1              
   
280
                 
   
84
)   280   ( 3      
          252            
          28 )   84   ( 3
                84      
                0      

As last divisor is 28, then HCF is 28.

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Three different containers contain 144 liters, 192 liters and 288 liters of petrol respectively. Which biggest measure can measure all these different quantities exactly ?
  • 46 liters
  • 72 liters
  • 96 liters
  • 63 liters
  • 64 liters
Explanation   

HCF of 144, 192 and 288 is


192 )
288
( 1              
   
192
                 
   
96
) 192 ( 2          
        192              
        0              

0 =>  HCF of 192 and 288 is 96, again HCF of 144 and 192 is


144 )
192
( 1              
    144                  
   
48
) 144 ( 3          
        144              
        0              

HCF of 144, 192 and 288 is 46, 46 liter is the biggest measure, that can be used.

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Three numbers are in the ratio 2:3:5 respectively, and their HCF is 15. What are the numbers ?
  • 45, 75, 30
  • 30, 45, 75
  • 30, 75, 45
  • 15, 30, 45
  • 45, 60, 90
Explanation   

Let the numbers be 2a: 3a: 5a, then HCF of the numbers would be ‘a’

Therefore the numbers are 2 x 15 = 30, 3 x 15 = 45, 5 x 15 = 75

Workspace
What is the largest number that will divide 109, 206 and 303 leaving the same remainder in each case ?
  • 32
  • 81
  • 63
  • 97
  • 67
Explanation   

Greatest number that can divide x, y and z, leaving the same remainder in each case is

HCF of ( x – y ), ( y – z ) ( z – x ) [ i.e. the difference between the given numbers]

( 303 – 206) = 97, ( 303 – 109 ) = 194, (206 – 109 ) = 97.

HCF of 97 is 97 and 194 is 97 = > Hence answer is 97

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What is the largest size of tile that can be paved in a room measuring 18 feet breadth and 24 feet length ?
  • 12 feet
  • 4 feet
  • 9 feet
  • 3 feet
  • 6 feet
Explanation   

HCF of 24 and 18 is 6 or 6 feet

Therefore answer is 6 feet.

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The sum of two numbers is 252 and their HCF is 36. What are the numbers ?
  • 78, 174
  • 92, 160
  • 66, 186
  • 108, 144
  • 87, 165
Explanation   

Let the number be 36a + 36b

$$ Then \ a \ + \ b \ = \ \frac{252}{36} \ = \ 7 $$

Co – Primes which sums up 7 are => (1,6) (2,5) (3,4)

The numbers can be (36, 216) or (72, 180) or (108, 144)

Out of these we have 108, 144 in the choices.

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Find LCM of 12, 18 and 26
  • 336
  • 468
  • 624
  • 182
  • 268
Explanation   

LCM of 12, 18 and 26 is

$$ \begin{array}{c|lcr} 2 & \text{12} & \text{18} & \text{26} \\ \hline 2 & \text{6} & \text{9} & \text{13} \\ \hline 3 & \text{3} & \text{9} & \text{13} \\ \hline & \text{1} & \text{3} & \text{13} \\ \end{array} $$

= > 22 x 32 x 13 or 4 x 9 x 13 = 468

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What is the least number which when divided by 24, 32 and 48, leaves 7 as its remainder ?
  • 55
  • 39
  • 113
  • 65
  • 105
Explanation   

LCM of 24, 32 and 48 is 96.

Required number = 96 + 7 = 113.

Workspace
What is the largest four digit number which is divisible by each one of 12, 14, 15 and 18 ?
  • 9620
  • 9720
  • 9820
  • 8820
  • 8920
Explanation   

LCM of 12, 14, 15 and 18 is 1260.

Largest 4 digit number divisible by 1260 is 8820 { Multiple of 1260 }

Workspace
What is the least number which when divided by 16, 18, 22 and 27 leaves the remainders 8, 10, 14 and 19 respectively ?
  • 4760
  • 4744
  • 4752
  • 4764
  • 4780
Explanation   

( 16 – 8 ) = 8, ( 18 – 10 ) = 8, ( 22 – 14 ) = 8, (27 – 19 ) = 8

LCM of 16, 18, 22 and 27 is 4752

Required number is 4752 – 8 = 4744

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Quantitative Aptitude H.C.F and L.C.M Important Formulas

Highest Common Factor / Greatest Common Measure [H.C.F/G.C.M]: The H.C.F. of two or more than two numbers is the greatest number that divides each one of them exactly.

H.C.F by Factorization:

Express each one of the given numbers as the product of prime factors. The product of least powers of common prime factors gives HCF.

H.C.F. by Division Method:

  • To find the HCF of two given numbers divide the larger number by the smallest.
  • Divide the divisor by the remainder. Repeat the process of dividing by the preceding divisor by the remainder last obtained till Zero is obtained as remainder.
  • The last divisor is the required HCF.

Lowest Common Multiple [L.C.M]

The least number which is exactly divisible by each one of the given numbers is called their LCM.

LCM by Factorization Method:

Express each one of the given numbers as the product of prime factors. The product of highest powers of all prime factors gives LCM.

  • Product of two numbers =  Product of their HCF & LCM

H.C.F & L.C.M of Fractions:

HCF = $\displaystyle \frac{\text{HCF of Numerators}}{\text {LCM of Denominators}}$

LCM = $\displaystyle \frac{\text{LCM of Numerators}}{\text{HCF of Denominators}}$

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