HCF & LCM

Important Facts And Formulae

1. FACTORS AND MULTIPLES

If a divides b exactly, then we say that a is a factor of b and that b is a multiple of a.

Ex. (i) 2 is a factor of 8—-> 8 is a multiple of 2.

(ii) 5 is a factor of 15 —> 15 is a multiple of 5.

2. Product of two numbers =(Their H.C.F) \times (Their L.C.M)

3. Product of n numbers=(H.C.F) ^{n-1} \times L.C.M

4. CO-PRIMES : Two numbers are co-prime if their H.C.F is 1.

5. H.C.F AND L.C.M OF FRACTIONS

(i) H.C.F = \frac{H.C.F of numerators}{L.C.M of denominators}

(ii) L.C.M = \frac{L.C.M of numerators}{H.C.F of denominators}

 

 

ILLUSTRATIVE EXAMPLES

1. The product of two numbers is 1280 and their H.C.F. is 8. The L.C.M of these numbers is:                                      (SSC. 2007)

Answer: Let the LCM be x. Then,

(H.C.F) \times  (L.C.M) = product of two numbers

Therefore, L.C.M = \frac{1280}{8} = 160

2. The L.C.M of two numbers is 12 times their H.C.F. The sum of H.C.F and L.C.M is 403. If one of the numbers is 93, then the other number is: (SSC. 2008)

Answer: Let H.C.F be x. Then, L.C.M=12x

Therefore, x + 12x = 403

\Rightarrow 13x  = 403 \Rightarrow x= 31

Therefore, H.C.F = 31 and L.C.M = (12 \times 31) = 372

Let the other number be y,

Then, 31 \times 372 = 93 \times y

\Rightarrow y = \frac{31\times372}{93}= 124

Therefore, Other number = 124

3. The product of two numbers is 4107 and their H.C.F is 37. The larger number is: (SSC. 2008)

Answer: Let the number be 37 a and 37 b, where a and b are co-primes.

Then, 37 a \times 37 b = 4107

\Rightarrow ab = \frac{4107}{37\times37} = 3.

Let us take a = 1, b= 3. Then,

The numbers are 37 \times 1 and 37 \times 3 i.e. 37 and 111.

Hence, the larger number is 111.

4. The number nearest to 43582 divisible by each of 25, 50, 75 is :  (SSC. 2007)

Answer:   We have 

  

L.C.M. of 25, 50, 75 = (25 \times 2 \times 3 ) = 150

On dividing 43582 by 150 we get 82 as remainder.

Therefore, Required number = (43582 – 82) = 43500

5. The H.C.F of two numbers, each having 3-digits is 17 and their L.C.M is 714. The sum of the numbers is :  (SSC. 2007)

Answer: Let the two numbers be 17 a and 17 b, where a and b are co-primes. Then,

L.C.M = 17ab

Therefore, 17 ab =714  \Rightarrow ab = \frac{714}{17} = 42

The possible pairs of co-primes are (1, 42) , (2, 21) , (3 , 14) , (6, 7).

Possible pairs of numbers are:

(17 \times 1,17 \times 42),   (17 \times 2, 17 \times 21),   (17 \times 3, 17 \times 14) ,   (17 \times 6, 17 \times 7)

Out of these the 3-digit numbers are 102 and 119 only. Their sum is 221.

6. The H.C.F of two numbers is 15 and their product is 6300. How many such pairs of  numbers are there ?   (SSC. 2008)

Answer: Let the number be 15 a and 15 b , where a and b are co-primes.

Therefore, 15 a \times 15 b = 6300

\Rightarrow ab = \frac{6300}{15\times15} = 28

Now, the pairs of two numbers which are relatively co-prime and whose product is 28 are (1, 28) and (4, 7) only.

Thus, 2 such pairs exist.

7. The H.C.F and L.C.M of two numbers are 21 and 4641 respectively. If one of the numbers lies between 200 and 300, then the two numbers are: ?   (M.B.A. 2006)

Answer: Let the number be 21 a and 21 b, where a and b are co-primes. Then, (21 a \times 21 b) = (21 \times 4641)\Rightarrow ab=221

Two co-primes with product 221 are 13 and 17.

Therefore, The numbers are (21 \times 13, 21 \times 17 ) i.e. 273 and 357

8. The number nearest to 10000, which is exactly divisible by each of the numbers 3,4,5,6,7,8 is:    (S.S.C. 2004)

Answer: L.C.M of 3, 4, 5, 6, 7, 8 = (2 \times  2 \times 3 \times  5 \times  7 \times 2) = 840

On dividing 10000 by 840, we get quotient = 11 and remainder = 760

Therefore, Required number will be [10000 + (840-760)]

i.e., Nearest number is 10080

CLICK HERE to Learn Previous Year Asked Important Questions on LCM and HCF

 

 

 

 

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