Maths - LCM and HCF

Prime factorisation:If a natural number is expressed as the product of prime numbers, then the factorisation of the number is called its prime factorisation.

For example: (i) 24 = 2×2×2×3 = 2³×3, (ii) 60 = 2×2×3×5 = 2² ×3×5.
Least Common Multiple (L.C.M.) of two natural numbers is the smallest natural number which is a multiple of both the numbers.
Highest Common Factor (H.C.F.) of two natural numbers is the largest common factor (or divisor) of the given natural numbers. In other words, H.C.F. is the greatest factor of the given numbers. H.C.F. is also called Greatest Common Divisor (G.C.D.)
Relation between L.C.M. and H.C.F. of two natural number: No 1 x No 2 = Their L.C.M. x H.C.F
Example: Find the L.C.M. of 72, 240, 196.
Using Prime factorisation method
   72 = 2×2×2×3×3 = 2³×3²,      240 = 2×2×2×2×3×5 = 2⁴×3×5
   196 = 2×2×7×7 = 2²×7²
L.C.M. of the given numbers = product of all the prime factors of each of the given number with greatest index of common prime factors
     = 2⁴×3²×5×7² = 16×9×5×49 = 35280.
Using Division method
  2 | 72, 240, 196
  2 | 36, 120, 98
   2 | 18, 60 , 49
  3 | 9 , 30 , 49
      | 3  , 10 , 49
 L.C.M. of the given numbers
    = product of divisors and the remaining numbers
    = 2×2×2×3×3×10×49
    = 72×10×49 = 35280.

Example: Find the H.C.F. of 72, 126 and 270.

Using Prime factorisation method
    72 = 2×2×2×3×3 = 2³×3²,      126 = 2×3×3×7 = 2¹×3²×7¹
  270 = 2×3×3×3×5 = 2¹×3³×5¹
H.C.F. of the given numbers = the product of common factors with least index
  = 2¹×3² = 2×3×3 = 18
Using Division method
First find H.C.F. of 72 and 126
72|126|1
      72       
       54| 72|1
              54
              18| 54| 3
                     54
                      0 
 H.C.F. of 72 and 126 = 18
Similarly calculate H.C.F. of 18 and 270 as 18
Hence H.C.F. of the given three numbers = 18

Exercise

1. 1095/1168 when expressed in simplest form is :

      a. 13/16 b. 15/16 C. 17/26 d. 25/26 e. None

2. HCF of 4 x 27 x 3125, 8 x 9 25 x 7 and 16 x 81 x 5 x 11 x 49 :

      a. 180 b. 360 c. 540 d. 1260 e. None

3. Which of the following is co-prime:

      a. 16,62 b. 18,25 c. 21,35 d. 23,92 e. None

4. Find the lowest common multiple of 24, 36 and 40 :

       a. 120 b. 240 c. 360 d. 480 e. None

5. Three numbers are in the ratio of 1:2:3 and their HCF is 12. numbers are:

       a. 4, 8, 12 b. 5, 10, 15 c. 10, 20, 30 d. 12, 24, 36 e. None

6. The LCM of two no is 48, and are in ratio of 2:3. The sum of no is:

      a. 28 b. 32 c. 40 d. 64 e. None

7. LCM of two no is 495 and HCF is 5. If their sum is 100, then diffrence is :

      a. 10 b. 46 c. 70 d. 90 e. None

8. The HCF of two numbers is 8, which one of following can never be their LCM:

      a. 24 b. 48 c. 56 d. 60 e. None

9. The greatest number that exactly divides 105, 1001 and 2436 is:

      a. 3 b. 7 c. 11 d. 21 e. None

10. The greatest possible length which can be used to measure exactly 7m, 3m 85cm and 12m 95cm:

      a. 15 cm b. 25 cm c. 35 cm d. 42 cm e. None

11. Find the greatest number that will divide 43, 91 & 183 so as to leave same remainder:

      a. 4 b. 7 c. 9 d. 13 e. None

12. The greatest number of four digit which is divisible by 15, 25, 40 and 75:

      a. 9000 b. 9400 c. 9600 d 9800 e. None

13. The least number which should be added to 2497 so that the sum is exactly divisible by 5, 6, 4 and 3:

      a. 3 b. 13 c. 23 d. 33 e. None

14. The smallest number whcih when diminished by 7, is divisible by 12, 16, 18, 21 and 28:

      a. 1008 b. 1015 c. 1022 d. 1032 e. None

15. The least number, which when divided by 12, 15, 20 and 54 leaves in each case a reminder of 8 is:

      a. 504 b. 536 c. 544 d. 548 e. None

16. The HCF of 0.54, 1.8 and 7.2 is:

      a. 1.8 b. 0.18 c. 0.018 d. 18 e. None 
Answer: 1b, 2a, 3b, 4c, 5d, 6c, 7a, 8d, 9b, 10c, 11a, 12c, 13c, 14b, 15d, 16b